Integral de x^4/(2*x^4+11*x^2+13)^2 dx
Solución
Solución detallada
Hay varias maneras de calcular esta integral.
Método #1
Vuelva a escribir el integrando:
x 4 ( ( 2 x 4 + 11 x 2 ) + 13 ) 2 = − 11 x 2 + 13 2 ( 2 x 4 + 11 x 2 + 13 ) 2 + 1 2 ( 2 x 4 + 11 x 2 + 13 ) \frac{x^{4}}{\left(\left(2 x^{4} + 11 x^{2}\right) + 13\right)^{2}} = - \frac{11 x^{2} + 13}{2 \left(2 x^{4} + 11 x^{2} + 13\right)^{2}} + \frac{1}{2 \left(2 x^{4} + 11 x^{2} + 13\right)} ( ( 2 x 4 + 11 x 2 ) + 13 ) 2 x 4 = − 2 ( 2 x 4 + 11 x 2 + 13 ) 2 11 x 2 + 13 + 2 ( 2 x 4 + 11 x 2 + 13 ) 1
Integramos término a término:
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ ( − 11 x 2 + 13 2 ( 2 x 4 + 11 x 2 + 13 ) 2 ) d x = − ∫ 11 x 2 + 13 ( 2 x 4 + 11 x 2 + 13 ) 2 d x 2 \int \left(- \frac{11 x^{2} + 13}{2 \left(2 x^{4} + 11 x^{2} + 13\right)^{2}}\right)\, dx = - \frac{\int \frac{11 x^{2} + 13}{\left(2 x^{4} + 11 x^{2} + 13\right)^{2}}\, dx}{2} ∫ ( − 2 ( 2 x 4 + 11 x 2 + 13 ) 2 11 x 2 + 13 ) d x = − 2 ∫ ( 2 x 4 + 11 x 2 + 13 ) 2 11 x 2 + 13 d x
Vuelva a escribir el integrando:
11 x 2 + 13 ( 2 x 4 + 11 x 2 + 13 ) 2 = 11 x 2 + 13 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 \frac{11 x^{2} + 13}{\left(2 x^{4} + 11 x^{2} + 13\right)^{2}} = \frac{11 x^{2} + 13}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169} ( 2 x 4 + 11 x 2 + 13 ) 2 11 x 2 + 13 = 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 11 x 2 + 13
Vuelva a escribir el integrando:
11 x 2 + 13 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 = 11 x 2 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 + 13 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 \frac{11 x^{2} + 13}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169} = \frac{11 x^{2}}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169} + \frac{13}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169} 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 11 x 2 + 13 = 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 11 x 2 + 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 13
Integramos término a término:
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ 11 x 2 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 d x = 11 ∫ x 2 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 d x \int \frac{11 x^{2}}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169}\, dx = 11 \int \frac{x^{2}}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169}\, dx ∫ 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 11 x 2 d x = 11 ∫ 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 x 2 d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
− 4 x 3 + 11 x 68 x 4 + 374 x 2 + 442 − 2 4763 2043808 − 17 120224 atan ( 1868 26 x 11 17 4763 − 17 17 + 225 4763 − 17 17 ) − 2 17 120224 + 4763 2043808 atan ( 1868 26 x − 225 17 17 + 4763 + 11 17 17 17 + 4763 ) - \frac{4 x^{3} + 11 x}{68 x^{4} + 374 x^{2} + 442} - 2 \sqrt{\frac{4763}{2043808} - \frac{\sqrt{17}}{120224}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{11 \sqrt{17} \sqrt{4763 - 17 \sqrt{17}} + 225 \sqrt{4763 - 17 \sqrt{17}}} \right)} - 2 \sqrt{\frac{\sqrt{17}}{120224} + \frac{4763}{2043808}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{- 225 \sqrt{17 \sqrt{17} + 4763} + 11 \sqrt{17} \sqrt{17 \sqrt{17} + 4763}} \right)} − 68 x 4 + 374 x 2 + 442 4 x 3 + 11 x − 2 2043808 4763 − 120224 17 atan ( 11 17 4763 − 17 17 + 225 4763 − 17 17 1868 26 x ) − 2 120224 17 + 2043808 4763 atan ( − 225 17 17 + 4763 + 11 17 17 17 + 4763 1868 26 x )
Por lo tanto, el resultado es: − 11 ( 4 x 3 + 11 x ) 68 x 4 + 374 x 2 + 442 − 22 4763 2043808 − 17 120224 atan ( 1868 26 x 11 17 4763 − 17 17 + 225 4763 − 17 17 ) − 22 17 120224 + 4763 2043808 atan ( 1868 26 x − 225 17 17 + 4763 + 11 17 17 17 + 4763 ) - \frac{11 \left(4 x^{3} + 11 x\right)}{68 x^{4} + 374 x^{2} + 442} - 22 \sqrt{\frac{4763}{2043808} - \frac{\sqrt{17}}{120224}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{11 \sqrt{17} \sqrt{4763 - 17 \sqrt{17}} + 225 \sqrt{4763 - 17 \sqrt{17}}} \right)} - 22 \sqrt{\frac{\sqrt{17}}{120224} + \frac{4763}{2043808}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{- 225 \sqrt{17 \sqrt{17} + 4763} + 11 \sqrt{17} \sqrt{17 \sqrt{17} + 4763}} \right)} − 68 x 4 + 374 x 2 + 442 11 ( 4 x 3 + 11 x ) − 22 2043808 4763 − 120224 17 atan ( 11 17 4763 − 17 17 + 225 4763 − 17 17 1868 26 x ) − 22 120224 17 + 2043808 4763 atan ( − 225 17 17 + 4763 + 11 17 17 17 + 4763 1868 26 x )
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ 13 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 d x = 13 ∫ 1 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 d x \int \frac{13}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169}\, dx = 13 \int \frac{1}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169}\, dx ∫ 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 13 d x = 13 ∫ 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 1 d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
22 x 3 + 69 x 884 x 4 + 4862 x 2 + 5746 + 2 113 17 20317856 + 88121 345403552 atan ( 34424 26 x 35 17 1921 17 + 88121 + 957 1921 17 + 88121 ) + 2 88121 345403552 − 113 17 20317856 atan ( 34424 26 x − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 ) \frac{22 x^{3} + 69 x}{884 x^{4} + 4862 x^{2} + 5746} + 2 \sqrt{\frac{113 \sqrt{17}}{20317856} + \frac{88121}{345403552}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{35 \sqrt{17} \sqrt{1921 \sqrt{17} + 88121} + 957 \sqrt{1921 \sqrt{17} + 88121}} \right)} + 2 \sqrt{\frac{88121}{345403552} - \frac{113 \sqrt{17}}{20317856}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{- 957 \sqrt{88121 - 1921 \sqrt{17}} + 35 \sqrt{17} \sqrt{88121 - 1921 \sqrt{17}}} \right)} 884 x 4 + 4862 x 2 + 5746 22 x 3 + 69 x + 2 20317856 113 17 + 345403552 88121 atan ( 35 17 1921 17 + 88121 + 957 1921 17 + 88121 34424 26 x ) + 2 345403552 88121 − 20317856 113 17 atan ( − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 34424 26 x )
Por lo tanto, el resultado es: 13 ( 22 x 3 + 69 x ) 884 x 4 + 4862 x 2 + 5746 + 26 113 17 20317856 + 88121 345403552 atan ( 34424 26 x 35 17 1921 17 + 88121 + 957 1921 17 + 88121 ) + 26 88121 345403552 − 113 17 20317856 atan ( 34424 26 x − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 ) \frac{13 \left(22 x^{3} + 69 x\right)}{884 x^{4} + 4862 x^{2} + 5746} + 26 \sqrt{\frac{113 \sqrt{17}}{20317856} + \frac{88121}{345403552}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{35 \sqrt{17} \sqrt{1921 \sqrt{17} + 88121} + 957 \sqrt{1921 \sqrt{17} + 88121}} \right)} + 26 \sqrt{\frac{88121}{345403552} - \frac{113 \sqrt{17}}{20317856}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{- 957 \sqrt{88121 - 1921 \sqrt{17}} + 35 \sqrt{17} \sqrt{88121 - 1921 \sqrt{17}}} \right)} 884 x 4 + 4862 x 2 + 5746 13 ( 22 x 3 + 69 x ) + 26 20317856 113 17 + 345403552 88121 atan ( 35 17 1921 17 + 88121 + 957 1921 17 + 88121 34424 26 x ) + 26 345403552 88121 − 20317856 113 17 atan ( − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 34424 26 x )
El resultado es: − 11 ( 4 x 3 + 11 x ) 68 x 4 + 374 x 2 + 442 + 13 ( 22 x 3 + 69 x ) 884 x 4 + 4862 x 2 + 5746 − 22 4763 2043808 − 17 120224 atan ( 1868 26 x 11 17 4763 − 17 17 + 225 4763 − 17 17 ) + 26 113 17 20317856 + 88121 345403552 atan ( 34424 26 x 35 17 1921 17 + 88121 + 957 1921 17 + 88121 ) + 26 88121 345403552 − 113 17 20317856 atan ( 34424 26 x − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 ) − 22 17 120224 + 4763 2043808 atan ( 1868 26 x − 225 17 17 + 4763 + 11 17 17 17 + 4763 ) - \frac{11 \left(4 x^{3} + 11 x\right)}{68 x^{4} + 374 x^{2} + 442} + \frac{13 \left(22 x^{3} + 69 x\right)}{884 x^{4} + 4862 x^{2} + 5746} - 22 \sqrt{\frac{4763}{2043808} - \frac{\sqrt{17}}{120224}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{11 \sqrt{17} \sqrt{4763 - 17 \sqrt{17}} + 225 \sqrt{4763 - 17 \sqrt{17}}} \right)} + 26 \sqrt{\frac{113 \sqrt{17}}{20317856} + \frac{88121}{345403552}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{35 \sqrt{17} \sqrt{1921 \sqrt{17} + 88121} + 957 \sqrt{1921 \sqrt{17} + 88121}} \right)} + 26 \sqrt{\frac{88121}{345403552} - \frac{113 \sqrt{17}}{20317856}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{- 957 \sqrt{88121 - 1921 \sqrt{17}} + 35 \sqrt{17} \sqrt{88121 - 1921 \sqrt{17}}} \right)} - 22 \sqrt{\frac{\sqrt{17}}{120224} + \frac{4763}{2043808}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{- 225 \sqrt{17 \sqrt{17} + 4763} + 11 \sqrt{17} \sqrt{17 \sqrt{17} + 4763}} \right)} − 68 x 4 + 374 x 2 + 442 11 ( 4 x 3 + 11 x ) + 884 x 4 + 4862 x 2 + 5746 13 ( 22 x 3 + 69 x ) − 22 2043808 4763 − 120224 17 atan ( 11 17 4763 − 17 17 + 225 4763 − 17 17 1868 26 x ) + 26 20317856 113 17 + 345403552 88121 atan ( 35 17 1921 17 + 88121 + 957 1921 17 + 88121 34424 26 x ) + 26 345403552 88121 − 20317856 113 17 atan ( − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 34424 26 x ) − 22 120224 17 + 2043808 4763 atan ( − 225 17 17 + 4763 + 11 17 17 17 + 4763 1868 26 x )
Por lo tanto, el resultado es: 11 ( 4 x 3 + 11 x ) 2 ( 68 x 4 + 374 x 2 + 442 ) − 13 ( 22 x 3 + 69 x ) 2 ( 884 x 4 + 4862 x 2 + 5746 ) + 11 4763 2043808 − 17 120224 atan ( 1868 26 x 11 17 4763 − 17 17 + 225 4763 − 17 17 ) − 13 113 17 20317856 + 88121 345403552 atan ( 34424 26 x 35 17 1921 17 + 88121 + 957 1921 17 + 88121 ) − 13 88121 345403552 − 113 17 20317856 atan ( 34424 26 x − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 ) + 11 17 120224 + 4763 2043808 atan ( 1868 26 x − 225 17 17 + 4763 + 11 17 17 17 + 4763 ) \frac{11 \left(4 x^{3} + 11 x\right)}{2 \left(68 x^{4} + 374 x^{2} + 442\right)} - \frac{13 \left(22 x^{3} + 69 x\right)}{2 \left(884 x^{4} + 4862 x^{2} + 5746\right)} + 11 \sqrt{\frac{4763}{2043808} - \frac{\sqrt{17}}{120224}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{11 \sqrt{17} \sqrt{4763 - 17 \sqrt{17}} + 225 \sqrt{4763 - 17 \sqrt{17}}} \right)} - 13 \sqrt{\frac{113 \sqrt{17}}{20317856} + \frac{88121}{345403552}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{35 \sqrt{17} \sqrt{1921 \sqrt{17} + 88121} + 957 \sqrt{1921 \sqrt{17} + 88121}} \right)} - 13 \sqrt{\frac{88121}{345403552} - \frac{113 \sqrt{17}}{20317856}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{- 957 \sqrt{88121 - 1921 \sqrt{17}} + 35 \sqrt{17} \sqrt{88121 - 1921 \sqrt{17}}} \right)} + 11 \sqrt{\frac{\sqrt{17}}{120224} + \frac{4763}{2043808}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{- 225 \sqrt{17 \sqrt{17} + 4763} + 11 \sqrt{17} \sqrt{17 \sqrt{17} + 4763}} \right)} 2 ( 68 x 4 + 374 x 2 + 442 ) 11 ( 4 x 3 + 11 x ) − 2 ( 884 x 4 + 4862 x 2 + 5746 ) 13 ( 22 x 3 + 69 x ) + 11 2043808 4763 − 120224 17 atan ( 11 17 4763 − 17 17 + 225 4763 − 17 17 1868 26 x ) − 13 20317856 113 17 + 345403552 88121 atan ( 35 17 1921 17 + 88121 + 957 1921 17 + 88121 34424 26 x ) − 13 345403552 88121 − 20317856 113 17 atan ( − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 34424 26 x ) + 11 120224 17 + 2043808 4763 atan ( − 225 17 17 + 4763 + 11 17 17 17 + 4763 1868 26 x )
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ 1 2 ( 2 x 4 + 11 x 2 + 13 ) d x = ∫ 1 2 x 4 + 11 x 2 + 13 d x 2 \int \frac{1}{2 \left(2 x^{4} + 11 x^{2} + 13\right)}\, dx = \frac{\int \frac{1}{2 x^{4} + 11 x^{2} + 13}\, dx}{2} ∫ 2 ( 2 x 4 + 11 x 2 + 13 ) 1 d x = 2 ∫ 2 x 4 + 11 x 2 + 13 1 d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
− 2 11 1768 − 17 1768 atan ( 4 26 x 17 11 − 17 + 11 11 − 17 ) − 2 17 1768 + 11 1768 atan ( 4 26 x − 11 17 + 11 + 17 17 + 11 ) - 2 \sqrt{\frac{11}{1768} - \frac{\sqrt{17}}{1768}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{\sqrt{17} \sqrt{11 - \sqrt{17}} + 11 \sqrt{11 - \sqrt{17}}} \right)} - 2 \sqrt{\frac{\sqrt{17}}{1768} + \frac{11}{1768}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{- 11 \sqrt{\sqrt{17} + 11} + \sqrt{17} \sqrt{\sqrt{17} + 11}} \right)} − 2 1768 11 − 1768 17 atan ( 17 11 − 17 + 11 11 − 17 4 26 x ) − 2 1768 17 + 1768 11 atan ( − 11 17 + 11 + 17 17 + 11 4 26 x )
Por lo tanto, el resultado es: − 11 1768 − 17 1768 atan ( 4 26 x 17 11 − 17 + 11 11 − 17 ) − 17 1768 + 11 1768 atan ( 4 26 x − 11 17 + 11 + 17 17 + 11 ) - \sqrt{\frac{11}{1768} - \frac{\sqrt{17}}{1768}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{\sqrt{17} \sqrt{11 - \sqrt{17}} + 11 \sqrt{11 - \sqrt{17}}} \right)} - \sqrt{\frac{\sqrt{17}}{1768} + \frac{11}{1768}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{- 11 \sqrt{\sqrt{17} + 11} + \sqrt{17} \sqrt{\sqrt{17} + 11}} \right)} − 1768 11 − 1768 17 atan ( 17 11 − 17 + 11 11 − 17 4 26 x ) − 1768 17 + 1768 11 atan ( − 11 17 + 11 + 17 17 + 11 4 26 x )
El resultado es: 11 ( 4 x 3 + 11 x ) 2 ( 68 x 4 + 374 x 2 + 442 ) − 13 ( 22 x 3 + 69 x ) 2 ( 884 x 4 + 4862 x 2 + 5746 ) − 11 1768 − 17 1768 atan ( 4 26 x 17 11 − 17 + 11 11 − 17 ) + 11 4763 2043808 − 17 120224 atan ( 1868 26 x 11 17 4763 − 17 17 + 225 4763 − 17 17 ) − 13 113 17 20317856 + 88121 345403552 atan ( 34424 26 x 35 17 1921 17 + 88121 + 957 1921 17 + 88121 ) − 13 88121 345403552 − 113 17 20317856 atan ( 34424 26 x − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 ) − 17 1768 + 11 1768 atan ( 4 26 x − 11 17 + 11 + 17 17 + 11 ) + 11 17 120224 + 4763 2043808 atan ( 1868 26 x − 225 17 17 + 4763 + 11 17 17 17 + 4763 ) \frac{11 \left(4 x^{3} + 11 x\right)}{2 \left(68 x^{4} + 374 x^{2} + 442\right)} - \frac{13 \left(22 x^{3} + 69 x\right)}{2 \left(884 x^{4} + 4862 x^{2} + 5746\right)} - \sqrt{\frac{11}{1768} - \frac{\sqrt{17}}{1768}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{\sqrt{17} \sqrt{11 - \sqrt{17}} + 11 \sqrt{11 - \sqrt{17}}} \right)} + 11 \sqrt{\frac{4763}{2043808} - \frac{\sqrt{17}}{120224}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{11 \sqrt{17} \sqrt{4763 - 17 \sqrt{17}} + 225 \sqrt{4763 - 17 \sqrt{17}}} \right)} - 13 \sqrt{\frac{113 \sqrt{17}}{20317856} + \frac{88121}{345403552}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{35 \sqrt{17} \sqrt{1921 \sqrt{17} + 88121} + 957 \sqrt{1921 \sqrt{17} + 88121}} \right)} - 13 \sqrt{\frac{88121}{345403552} - \frac{113 \sqrt{17}}{20317856}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{- 957 \sqrt{88121 - 1921 \sqrt{17}} + 35 \sqrt{17} \sqrt{88121 - 1921 \sqrt{17}}} \right)} - \sqrt{\frac{\sqrt{17}}{1768} + \frac{11}{1768}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{- 11 \sqrt{\sqrt{17} + 11} + \sqrt{17} \sqrt{\sqrt{17} + 11}} \right)} + 11 \sqrt{\frac{\sqrt{17}}{120224} + \frac{4763}{2043808}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{- 225 \sqrt{17 \sqrt{17} + 4763} + 11 \sqrt{17} \sqrt{17 \sqrt{17} + 4763}} \right)} 2 ( 68 x 4 + 374 x 2 + 442 ) 11 ( 4 x 3 + 11 x ) − 2 ( 884 x 4 + 4862 x 2 + 5746 ) 13 ( 22 x 3 + 69 x ) − 1768 11 − 1768 17 atan ( 17 11 − 17 + 11 11 − 17 4 26 x ) + 11 2043808 4763 − 120224 17 atan ( 11 17 4763 − 17 17 + 225 4763 − 17 17 1868 26 x ) − 13 20317856 113 17 + 345403552 88121 atan ( 35 17 1921 17 + 88121 + 957 1921 17 + 88121 34424 26 x ) − 13 345403552 88121 − 20317856 113 17 atan ( − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 34424 26 x ) − 1768 17 + 1768 11 atan ( − 11 17 + 11 + 17 17 + 11 4 26 x ) + 11 120224 17 + 2043808 4763 atan ( − 225 17 17 + 4763 + 11 17 17 17 + 4763 1868 26 x )
Método #2
Vuelva a escribir el integrando:
x 4 ( ( 2 x 4 + 11 x 2 ) + 13 ) 2 = x 4 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 \frac{x^{4}}{\left(\left(2 x^{4} + 11 x^{2}\right) + 13\right)^{2}} = \frac{x^{4}}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169} ( ( 2 x 4 + 11 x 2 ) + 13 ) 2 x 4 = 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 x 4
Vuelva a escribir el integrando:
x 4 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 = − 11 x 2 + 13 2 ( 2 x 4 + 11 x 2 + 13 ) 2 + 1 2 ( 2 x 4 + 11 x 2 + 13 ) \frac{x^{4}}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169} = - \frac{11 x^{2} + 13}{2 \left(2 x^{4} + 11 x^{2} + 13\right)^{2}} + \frac{1}{2 \left(2 x^{4} + 11 x^{2} + 13\right)} 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 x 4 = − 2 ( 2 x 4 + 11 x 2 + 13 ) 2 11 x 2 + 13 + 2 ( 2 x 4 + 11 x 2 + 13 ) 1
Integramos término a término:
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ ( − 11 x 2 + 13 2 ( 2 x 4 + 11 x 2 + 13 ) 2 ) d x = − ∫ 11 x 2 + 13 ( 2 x 4 + 11 x 2 + 13 ) 2 d x 2 \int \left(- \frac{11 x^{2} + 13}{2 \left(2 x^{4} + 11 x^{2} + 13\right)^{2}}\right)\, dx = - \frac{\int \frac{11 x^{2} + 13}{\left(2 x^{4} + 11 x^{2} + 13\right)^{2}}\, dx}{2} ∫ ( − 2 ( 2 x 4 + 11 x 2 + 13 ) 2 11 x 2 + 13 ) d x = − 2 ∫ ( 2 x 4 + 11 x 2 + 13 ) 2 11 x 2 + 13 d x
Vuelva a escribir el integrando:
11 x 2 + 13 ( 2 x 4 + 11 x 2 + 13 ) 2 = 11 x 2 + 13 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 \frac{11 x^{2} + 13}{\left(2 x^{4} + 11 x^{2} + 13\right)^{2}} = \frac{11 x^{2} + 13}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169} ( 2 x 4 + 11 x 2 + 13 ) 2 11 x 2 + 13 = 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 11 x 2 + 13
Vuelva a escribir el integrando:
11 x 2 + 13 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 = 11 x 2 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 + 13 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 \frac{11 x^{2} + 13}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169} = \frac{11 x^{2}}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169} + \frac{13}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169} 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 11 x 2 + 13 = 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 11 x 2 + 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 13
Integramos término a término:
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ 11 x 2 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 d x = 11 ∫ x 2 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 d x \int \frac{11 x^{2}}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169}\, dx = 11 \int \frac{x^{2}}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169}\, dx ∫ 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 11 x 2 d x = 11 ∫ 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 x 2 d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
− 4 x 3 + 11 x 68 x 4 + 374 x 2 + 442 − 2 4763 2043808 − 17 120224 atan ( 1868 26 x 11 17 4763 − 17 17 + 225 4763 − 17 17 ) − 2 17 120224 + 4763 2043808 atan ( 1868 26 x − 225 17 17 + 4763 + 11 17 17 17 + 4763 ) - \frac{4 x^{3} + 11 x}{68 x^{4} + 374 x^{2} + 442} - 2 \sqrt{\frac{4763}{2043808} - \frac{\sqrt{17}}{120224}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{11 \sqrt{17} \sqrt{4763 - 17 \sqrt{17}} + 225 \sqrt{4763 - 17 \sqrt{17}}} \right)} - 2 \sqrt{\frac{\sqrt{17}}{120224} + \frac{4763}{2043808}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{- 225 \sqrt{17 \sqrt{17} + 4763} + 11 \sqrt{17} \sqrt{17 \sqrt{17} + 4763}} \right)} − 68 x 4 + 374 x 2 + 442 4 x 3 + 11 x − 2 2043808 4763 − 120224 17 atan ( 11 17 4763 − 17 17 + 225 4763 − 17 17 1868 26 x ) − 2 120224 17 + 2043808 4763 atan ( − 225 17 17 + 4763 + 11 17 17 17 + 4763 1868 26 x )
Por lo tanto, el resultado es: − 11 ( 4 x 3 + 11 x ) 68 x 4 + 374 x 2 + 442 − 22 4763 2043808 − 17 120224 atan ( 1868 26 x 11 17 4763 − 17 17 + 225 4763 − 17 17 ) − 22 17 120224 + 4763 2043808 atan ( 1868 26 x − 225 17 17 + 4763 + 11 17 17 17 + 4763 ) - \frac{11 \left(4 x^{3} + 11 x\right)}{68 x^{4} + 374 x^{2} + 442} - 22 \sqrt{\frac{4763}{2043808} - \frac{\sqrt{17}}{120224}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{11 \sqrt{17} \sqrt{4763 - 17 \sqrt{17}} + 225 \sqrt{4763 - 17 \sqrt{17}}} \right)} - 22 \sqrt{\frac{\sqrt{17}}{120224} + \frac{4763}{2043808}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{- 225 \sqrt{17 \sqrt{17} + 4763} + 11 \sqrt{17} \sqrt{17 \sqrt{17} + 4763}} \right)} − 68 x 4 + 374 x 2 + 442 11 ( 4 x 3 + 11 x ) − 22 2043808 4763 − 120224 17 atan ( 11 17 4763 − 17 17 + 225 4763 − 17 17 1868 26 x ) − 22 120224 17 + 2043808 4763 atan ( − 225 17 17 + 4763 + 11 17 17 17 + 4763 1868 26 x )
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ 13 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 d x = 13 ∫ 1 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 d x \int \frac{13}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169}\, dx = 13 \int \frac{1}{4 x^{8} + 44 x^{6} + 173 x^{4} + 286 x^{2} + 169}\, dx ∫ 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 13 d x = 13 ∫ 4 x 8 + 44 x 6 + 173 x 4 + 286 x 2 + 169 1 d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
22 x 3 + 69 x 884 x 4 + 4862 x 2 + 5746 + 2 113 17 20317856 + 88121 345403552 atan ( 34424 26 x 35 17 1921 17 + 88121 + 957 1921 17 + 88121 ) + 2 88121 345403552 − 113 17 20317856 atan ( 34424 26 x − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 ) \frac{22 x^{3} + 69 x}{884 x^{4} + 4862 x^{2} + 5746} + 2 \sqrt{\frac{113 \sqrt{17}}{20317856} + \frac{88121}{345403552}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{35 \sqrt{17} \sqrt{1921 \sqrt{17} + 88121} + 957 \sqrt{1921 \sqrt{17} + 88121}} \right)} + 2 \sqrt{\frac{88121}{345403552} - \frac{113 \sqrt{17}}{20317856}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{- 957 \sqrt{88121 - 1921 \sqrt{17}} + 35 \sqrt{17} \sqrt{88121 - 1921 \sqrt{17}}} \right)} 884 x 4 + 4862 x 2 + 5746 22 x 3 + 69 x + 2 20317856 113 17 + 345403552 88121 atan ( 35 17 1921 17 + 88121 + 957 1921 17 + 88121 34424 26 x ) + 2 345403552 88121 − 20317856 113 17 atan ( − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 34424 26 x )
Por lo tanto, el resultado es: 13 ( 22 x 3 + 69 x ) 884 x 4 + 4862 x 2 + 5746 + 26 113 17 20317856 + 88121 345403552 atan ( 34424 26 x 35 17 1921 17 + 88121 + 957 1921 17 + 88121 ) + 26 88121 345403552 − 113 17 20317856 atan ( 34424 26 x − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 ) \frac{13 \left(22 x^{3} + 69 x\right)}{884 x^{4} + 4862 x^{2} + 5746} + 26 \sqrt{\frac{113 \sqrt{17}}{20317856} + \frac{88121}{345403552}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{35 \sqrt{17} \sqrt{1921 \sqrt{17} + 88121} + 957 \sqrt{1921 \sqrt{17} + 88121}} \right)} + 26 \sqrt{\frac{88121}{345403552} - \frac{113 \sqrt{17}}{20317856}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{- 957 \sqrt{88121 - 1921 \sqrt{17}} + 35 \sqrt{17} \sqrt{88121 - 1921 \sqrt{17}}} \right)} 884 x 4 + 4862 x 2 + 5746 13 ( 22 x 3 + 69 x ) + 26 20317856 113 17 + 345403552 88121 atan ( 35 17 1921 17 + 88121 + 957 1921 17 + 88121 34424 26 x ) + 26 345403552 88121 − 20317856 113 17 atan ( − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 34424 26 x )
El resultado es: − 11 ( 4 x 3 + 11 x ) 68 x 4 + 374 x 2 + 442 + 13 ( 22 x 3 + 69 x ) 884 x 4 + 4862 x 2 + 5746 − 22 4763 2043808 − 17 120224 atan ( 1868 26 x 11 17 4763 − 17 17 + 225 4763 − 17 17 ) + 26 113 17 20317856 + 88121 345403552 atan ( 34424 26 x 35 17 1921 17 + 88121 + 957 1921 17 + 88121 ) + 26 88121 345403552 − 113 17 20317856 atan ( 34424 26 x − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 ) − 22 17 120224 + 4763 2043808 atan ( 1868 26 x − 225 17 17 + 4763 + 11 17 17 17 + 4763 ) - \frac{11 \left(4 x^{3} + 11 x\right)}{68 x^{4} + 374 x^{2} + 442} + \frac{13 \left(22 x^{3} + 69 x\right)}{884 x^{4} + 4862 x^{2} + 5746} - 22 \sqrt{\frac{4763}{2043808} - \frac{\sqrt{17}}{120224}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{11 \sqrt{17} \sqrt{4763 - 17 \sqrt{17}} + 225 \sqrt{4763 - 17 \sqrt{17}}} \right)} + 26 \sqrt{\frac{113 \sqrt{17}}{20317856} + \frac{88121}{345403552}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{35 \sqrt{17} \sqrt{1921 \sqrt{17} + 88121} + 957 \sqrt{1921 \sqrt{17} + 88121}} \right)} + 26 \sqrt{\frac{88121}{345403552} - \frac{113 \sqrt{17}}{20317856}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{- 957 \sqrt{88121 - 1921 \sqrt{17}} + 35 \sqrt{17} \sqrt{88121 - 1921 \sqrt{17}}} \right)} - 22 \sqrt{\frac{\sqrt{17}}{120224} + \frac{4763}{2043808}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{- 225 \sqrt{17 \sqrt{17} + 4763} + 11 \sqrt{17} \sqrt{17 \sqrt{17} + 4763}} \right)} − 68 x 4 + 374 x 2 + 442 11 ( 4 x 3 + 11 x ) + 884 x 4 + 4862 x 2 + 5746 13 ( 22 x 3 + 69 x ) − 22 2043808 4763 − 120224 17 atan ( 11 17 4763 − 17 17 + 225 4763 − 17 17 1868 26 x ) + 26 20317856 113 17 + 345403552 88121 atan ( 35 17 1921 17 + 88121 + 957 1921 17 + 88121 34424 26 x ) + 26 345403552 88121 − 20317856 113 17 atan ( − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 34424 26 x ) − 22 120224 17 + 2043808 4763 atan ( − 225 17 17 + 4763 + 11 17 17 17 + 4763 1868 26 x )
Por lo tanto, el resultado es: 11 ( 4 x 3 + 11 x ) 2 ( 68 x 4 + 374 x 2 + 442 ) − 13 ( 22 x 3 + 69 x ) 2 ( 884 x 4 + 4862 x 2 + 5746 ) + 11 4763 2043808 − 17 120224 atan ( 1868 26 x 11 17 4763 − 17 17 + 225 4763 − 17 17 ) − 13 113 17 20317856 + 88121 345403552 atan ( 34424 26 x 35 17 1921 17 + 88121 + 957 1921 17 + 88121 ) − 13 88121 345403552 − 113 17 20317856 atan ( 34424 26 x − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 ) + 11 17 120224 + 4763 2043808 atan ( 1868 26 x − 225 17 17 + 4763 + 11 17 17 17 + 4763 ) \frac{11 \left(4 x^{3} + 11 x\right)}{2 \left(68 x^{4} + 374 x^{2} + 442\right)} - \frac{13 \left(22 x^{3} + 69 x\right)}{2 \left(884 x^{4} + 4862 x^{2} + 5746\right)} + 11 \sqrt{\frac{4763}{2043808} - \frac{\sqrt{17}}{120224}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{11 \sqrt{17} \sqrt{4763 - 17 \sqrt{17}} + 225 \sqrt{4763 - 17 \sqrt{17}}} \right)} - 13 \sqrt{\frac{113 \sqrt{17}}{20317856} + \frac{88121}{345403552}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{35 \sqrt{17} \sqrt{1921 \sqrt{17} + 88121} + 957 \sqrt{1921 \sqrt{17} + 88121}} \right)} - 13 \sqrt{\frac{88121}{345403552} - \frac{113 \sqrt{17}}{20317856}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{- 957 \sqrt{88121 - 1921 \sqrt{17}} + 35 \sqrt{17} \sqrt{88121 - 1921 \sqrt{17}}} \right)} + 11 \sqrt{\frac{\sqrt{17}}{120224} + \frac{4763}{2043808}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{- 225 \sqrt{17 \sqrt{17} + 4763} + 11 \sqrt{17} \sqrt{17 \sqrt{17} + 4763}} \right)} 2 ( 68 x 4 + 374 x 2 + 442 ) 11 ( 4 x 3 + 11 x ) − 2 ( 884 x 4 + 4862 x 2 + 5746 ) 13 ( 22 x 3 + 69 x ) + 11 2043808 4763 − 120224 17 atan ( 11 17 4763 − 17 17 + 225 4763 − 17 17 1868 26 x ) − 13 20317856 113 17 + 345403552 88121 atan ( 35 17 1921 17 + 88121 + 957 1921 17 + 88121 34424 26 x ) − 13 345403552 88121 − 20317856 113 17 atan ( − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 34424 26 x ) + 11 120224 17 + 2043808 4763 atan ( − 225 17 17 + 4763 + 11 17 17 17 + 4763 1868 26 x )
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ 1 2 ( 2 x 4 + 11 x 2 + 13 ) d x = ∫ 1 2 x 4 + 11 x 2 + 13 d x 2 \int \frac{1}{2 \left(2 x^{4} + 11 x^{2} + 13\right)}\, dx = \frac{\int \frac{1}{2 x^{4} + 11 x^{2} + 13}\, dx}{2} ∫ 2 ( 2 x 4 + 11 x 2 + 13 ) 1 d x = 2 ∫ 2 x 4 + 11 x 2 + 13 1 d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
− 2 11 1768 − 17 1768 atan ( 4 26 x 17 11 − 17 + 11 11 − 17 ) − 2 17 1768 + 11 1768 atan ( 4 26 x − 11 17 + 11 + 17 17 + 11 ) - 2 \sqrt{\frac{11}{1768} - \frac{\sqrt{17}}{1768}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{\sqrt{17} \sqrt{11 - \sqrt{17}} + 11 \sqrt{11 - \sqrt{17}}} \right)} - 2 \sqrt{\frac{\sqrt{17}}{1768} + \frac{11}{1768}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{- 11 \sqrt{\sqrt{17} + 11} + \sqrt{17} \sqrt{\sqrt{17} + 11}} \right)} − 2 1768 11 − 1768 17 atan ( 17 11 − 17 + 11 11 − 17 4 26 x ) − 2 1768 17 + 1768 11 atan ( − 11 17 + 11 + 17 17 + 11 4 26 x )
Por lo tanto, el resultado es: − 11 1768 − 17 1768 atan ( 4 26 x 17 11 − 17 + 11 11 − 17 ) − 17 1768 + 11 1768 atan ( 4 26 x − 11 17 + 11 + 17 17 + 11 ) - \sqrt{\frac{11}{1768} - \frac{\sqrt{17}}{1768}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{\sqrt{17} \sqrt{11 - \sqrt{17}} + 11 \sqrt{11 - \sqrt{17}}} \right)} - \sqrt{\frac{\sqrt{17}}{1768} + \frac{11}{1768}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{- 11 \sqrt{\sqrt{17} + 11} + \sqrt{17} \sqrt{\sqrt{17} + 11}} \right)} − 1768 11 − 1768 17 atan ( 17 11 − 17 + 11 11 − 17 4 26 x ) − 1768 17 + 1768 11 atan ( − 11 17 + 11 + 17 17 + 11 4 26 x )
El resultado es: 11 ( 4 x 3 + 11 x ) 2 ( 68 x 4 + 374 x 2 + 442 ) − 13 ( 22 x 3 + 69 x ) 2 ( 884 x 4 + 4862 x 2 + 5746 ) − 11 1768 − 17 1768 atan ( 4 26 x 17 11 − 17 + 11 11 − 17 ) + 11 4763 2043808 − 17 120224 atan ( 1868 26 x 11 17 4763 − 17 17 + 225 4763 − 17 17 ) − 13 113 17 20317856 + 88121 345403552 atan ( 34424 26 x 35 17 1921 17 + 88121 + 957 1921 17 + 88121 ) − 13 88121 345403552 − 113 17 20317856 atan ( 34424 26 x − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 ) − 17 1768 + 11 1768 atan ( 4 26 x − 11 17 + 11 + 17 17 + 11 ) + 11 17 120224 + 4763 2043808 atan ( 1868 26 x − 225 17 17 + 4763 + 11 17 17 17 + 4763 ) \frac{11 \left(4 x^{3} + 11 x\right)}{2 \left(68 x^{4} + 374 x^{2} + 442\right)} - \frac{13 \left(22 x^{3} + 69 x\right)}{2 \left(884 x^{4} + 4862 x^{2} + 5746\right)} - \sqrt{\frac{11}{1768} - \frac{\sqrt{17}}{1768}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{\sqrt{17} \sqrt{11 - \sqrt{17}} + 11 \sqrt{11 - \sqrt{17}}} \right)} + 11 \sqrt{\frac{4763}{2043808} - \frac{\sqrt{17}}{120224}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{11 \sqrt{17} \sqrt{4763 - 17 \sqrt{17}} + 225 \sqrt{4763 - 17 \sqrt{17}}} \right)} - 13 \sqrt{\frac{113 \sqrt{17}}{20317856} + \frac{88121}{345403552}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{35 \sqrt{17} \sqrt{1921 \sqrt{17} + 88121} + 957 \sqrt{1921 \sqrt{17} + 88121}} \right)} - 13 \sqrt{\frac{88121}{345403552} - \frac{113 \sqrt{17}}{20317856}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{- 957 \sqrt{88121 - 1921 \sqrt{17}} + 35 \sqrt{17} \sqrt{88121 - 1921 \sqrt{17}}} \right)} - \sqrt{\frac{\sqrt{17}}{1768} + \frac{11}{1768}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{- 11 \sqrt{\sqrt{17} + 11} + \sqrt{17} \sqrt{\sqrt{17} + 11}} \right)} + 11 \sqrt{\frac{\sqrt{17}}{120224} + \frac{4763}{2043808}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{- 225 \sqrt{17 \sqrt{17} + 4763} + 11 \sqrt{17} \sqrt{17 \sqrt{17} + 4763}} \right)} 2 ( 68 x 4 + 374 x 2 + 442 ) 11 ( 4 x 3 + 11 x ) − 2 ( 884 x 4 + 4862 x 2 + 5746 ) 13 ( 22 x 3 + 69 x ) − 1768 11 − 1768 17 atan ( 17 11 − 17 + 11 11 − 17 4 26 x ) + 11 2043808 4763 − 120224 17 atan ( 11 17 4763 − 17 17 + 225 4763 − 17 17 1868 26 x ) − 13 20317856 113 17 + 345403552 88121 atan ( 35 17 1921 17 + 88121 + 957 1921 17 + 88121 34424 26 x ) − 13 345403552 88121 − 20317856 113 17 atan ( − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 34424 26 x ) − 1768 17 + 1768 11 atan ( − 11 17 + 11 + 17 17 + 11 4 26 x ) + 11 120224 17 + 2043808 4763 atan ( − 225 17 17 + 4763 + 11 17 17 17 + 4763 1868 26 x )
Ahora simplificar:
4862 x ( 4 x 2 + 11 ) − 442 x ( 22 x 2 + 69 ) + 442 ( 2 x 4 + 11 x 2 + 13 ) ( − 88121 − 1921 17 atan ( 34424 26 x ( − 957 + 35 17 ) 88121 − 1921 17 ) + 11 17 17 + 4763 atan ( 1868 26 x ( − 225 + 11 17 ) 17 17 + 4763 ) − 34 17 + 11 atan ( 4 26 x ( − 11 + 17 ) 17 + 11 ) − 34 11 − 17 atan ( 4 26 x 11 − 17 ( 17 + 11 ) ) + 11 4763 − 17 17 atan ( 1868 26 x 4763 − 17 17 ( 11 17 + 225 ) ) − 1921 17 + 88121 atan ( 34424 26 x ( 35 17 + 957 ) 1921 17 + 88121 ) ) 30056 ( 2 x 4 + 11 x 2 + 13 ) \frac{4862 x \left(4 x^{2} + 11\right) - 442 x \left(22 x^{2} + 69\right) + \sqrt{442} \left(2 x^{4} + 11 x^{2} + 13\right) \left(- \sqrt{88121 - 1921 \sqrt{17}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{\left(-957 + 35 \sqrt{17}\right) \sqrt{88121 - 1921 \sqrt{17}}} \right)} + 11 \sqrt{17 \sqrt{17} + 4763} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{\left(-225 + 11 \sqrt{17}\right) \sqrt{17 \sqrt{17} + 4763}} \right)} - 34 \sqrt{\sqrt{17} + 11} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{\left(-11 + \sqrt{17}\right) \sqrt{\sqrt{17} + 11}} \right)} - 34 \sqrt{11 - \sqrt{17}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{\sqrt{11 - \sqrt{17}} \left(\sqrt{17} + 11\right)} \right)} + 11 \sqrt{4763 - 17 \sqrt{17}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{\sqrt{4763 - 17 \sqrt{17}} \left(11 \sqrt{17} + 225\right)} \right)} - \sqrt{1921 \sqrt{17} + 88121} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{\left(35 \sqrt{17} + 957\right) \sqrt{1921 \sqrt{17} + 88121}} \right)}\right)}{30056 \left(2 x^{4} + 11 x^{2} + 13\right)} 30056 ( 2 x 4 + 11 x 2 + 13 ) 4862 x ( 4 x 2 + 11 ) − 442 x ( 22 x 2 + 69 ) + 442 ( 2 x 4 + 11 x 2 + 13 ) ( − 88121 − 1921 17 atan ( ( − 957 + 35 17 ) 88121 − 1921 17 34424 26 x ) + 11 17 17 + 4763 atan ( ( − 225 + 11 17 ) 17 17 + 4763 1868 26 x ) − 34 17 + 11 atan ( ( − 11 + 17 ) 17 + 11 4 26 x ) − 34 11 − 17 atan ( 11 − 17 ( 17 + 11 ) 4 26 x ) + 11 4763 − 17 17 atan ( 4763 − 17 17 ( 11 17 + 225 ) 1868 26 x ) − 1921 17 + 88121 atan ( ( 35 17 + 957 ) 1921 17 + 88121 34424 26 x ) )
Añadimos la constante de integración:
4862 x ( 4 x 2 + 11 ) − 442 x ( 22 x 2 + 69 ) + 442 ( 2 x 4 + 11 x 2 + 13 ) ( − 88121 − 1921 17 atan ( 34424 26 x ( − 957 + 35 17 ) 88121 − 1921 17 ) + 11 17 17 + 4763 atan ( 1868 26 x ( − 225 + 11 17 ) 17 17 + 4763 ) − 34 17 + 11 atan ( 4 26 x ( − 11 + 17 ) 17 + 11 ) − 34 11 − 17 atan ( 4 26 x 11 − 17 ( 17 + 11 ) ) + 11 4763 − 17 17 atan ( 1868 26 x 4763 − 17 17 ( 11 17 + 225 ) ) − 1921 17 + 88121 atan ( 34424 26 x ( 35 17 + 957 ) 1921 17 + 88121 ) ) 30056 ( 2 x 4 + 11 x 2 + 13 ) + c o n s t a n t \frac{4862 x \left(4 x^{2} + 11\right) - 442 x \left(22 x^{2} + 69\right) + \sqrt{442} \left(2 x^{4} + 11 x^{2} + 13\right) \left(- \sqrt{88121 - 1921 \sqrt{17}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{\left(-957 + 35 \sqrt{17}\right) \sqrt{88121 - 1921 \sqrt{17}}} \right)} + 11 \sqrt{17 \sqrt{17} + 4763} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{\left(-225 + 11 \sqrt{17}\right) \sqrt{17 \sqrt{17} + 4763}} \right)} - 34 \sqrt{\sqrt{17} + 11} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{\left(-11 + \sqrt{17}\right) \sqrt{\sqrt{17} + 11}} \right)} - 34 \sqrt{11 - \sqrt{17}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{\sqrt{11 - \sqrt{17}} \left(\sqrt{17} + 11\right)} \right)} + 11 \sqrt{4763 - 17 \sqrt{17}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{\sqrt{4763 - 17 \sqrt{17}} \left(11 \sqrt{17} + 225\right)} \right)} - \sqrt{1921 \sqrt{17} + 88121} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{\left(35 \sqrt{17} + 957\right) \sqrt{1921 \sqrt{17} + 88121}} \right)}\right)}{30056 \left(2 x^{4} + 11 x^{2} + 13\right)}+ \mathrm{constant} 30056 ( 2 x 4 + 11 x 2 + 13 ) 4862 x ( 4 x 2 + 11 ) − 442 x ( 22 x 2 + 69 ) + 442 ( 2 x 4 + 11 x 2 + 13 ) ( − 88121 − 1921 17 atan ( ( − 957 + 35 17 ) 88121 − 1921 17 34424 26 x ) + 11 17 17 + 4763 atan ( ( − 225 + 11 17 ) 17 17 + 4763 1868 26 x ) − 34 17 + 11 atan ( ( − 11 + 17 ) 17 + 11 4 26 x ) − 34 11 − 17 atan ( 11 − 17 ( 17 + 11 ) 4 26 x ) + 11 4763 − 17 17 atan ( 4763 − 17 17 ( 11 17 + 225 ) 1868 26 x ) − 1921 17 + 88121 atan ( ( 35 17 + 957 ) 1921 17 + 88121 34424 26 x ) ) + constant
Respuesta:
4862 x ( 4 x 2 + 11 ) − 442 x ( 22 x 2 + 69 ) + 442 ( 2 x 4 + 11 x 2 + 13 ) ( − 88121 − 1921 17 atan ( 34424 26 x ( − 957 + 35 17 ) 88121 − 1921 17 ) + 11 17 17 + 4763 atan ( 1868 26 x ( − 225 + 11 17 ) 17 17 + 4763 ) − 34 17 + 11 atan ( 4 26 x ( − 11 + 17 ) 17 + 11 ) − 34 11 − 17 atan ( 4 26 x 11 − 17 ( 17 + 11 ) ) + 11 4763 − 17 17 atan ( 1868 26 x 4763 − 17 17 ( 11 17 + 225 ) ) − 1921 17 + 88121 atan ( 34424 26 x ( 35 17 + 957 ) 1921 17 + 88121 ) ) 30056 ( 2 x 4 + 11 x 2 + 13 ) + c o n s t a n t \frac{4862 x \left(4 x^{2} + 11\right) - 442 x \left(22 x^{2} + 69\right) + \sqrt{442} \left(2 x^{4} + 11 x^{2} + 13\right) \left(- \sqrt{88121 - 1921 \sqrt{17}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{\left(-957 + 35 \sqrt{17}\right) \sqrt{88121 - 1921 \sqrt{17}}} \right)} + 11 \sqrt{17 \sqrt{17} + 4763} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{\left(-225 + 11 \sqrt{17}\right) \sqrt{17 \sqrt{17} + 4763}} \right)} - 34 \sqrt{\sqrt{17} + 11} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{\left(-11 + \sqrt{17}\right) \sqrt{\sqrt{17} + 11}} \right)} - 34 \sqrt{11 - \sqrt{17}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{\sqrt{11 - \sqrt{17}} \left(\sqrt{17} + 11\right)} \right)} + 11 \sqrt{4763 - 17 \sqrt{17}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{\sqrt{4763 - 17 \sqrt{17}} \left(11 \sqrt{17} + 225\right)} \right)} - \sqrt{1921 \sqrt{17} + 88121} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{\left(35 \sqrt{17} + 957\right) \sqrt{1921 \sqrt{17} + 88121}} \right)}\right)}{30056 \left(2 x^{4} + 11 x^{2} + 13\right)}+ \mathrm{constant} 30056 ( 2 x 4 + 11 x 2 + 13 ) 4862 x ( 4 x 2 + 11 ) − 442 x ( 22 x 2 + 69 ) + 442 ( 2 x 4 + 11 x 2 + 13 ) ( − 88121 − 1921 17 atan ( ( − 957 + 35 17 ) 88121 − 1921 17 34424 26 x ) + 11 17 17 + 4763 atan ( ( − 225 + 11 17 ) 17 17 + 4763 1868 26 x ) − 34 17 + 11 atan ( ( − 11 + 17 ) 17 + 11 4 26 x ) − 34 11 − 17 atan ( 11 − 17 ( 17 + 11 ) 4 26 x ) + 11 4763 − 17 17 atan ( 4763 − 17 17 ( 11 17 + 225 ) 1868 26 x ) − 1921 17 + 88121 atan ( ( 35 17 + 957 ) 1921 17 + 88121 34424 26 x ) ) + constant
Respuesta (Indefinida)
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| _______________ _______________ ________________________ ________________________ __________________ __________________
| 4 / ____ / ____ \ / ____ / ____ \ / ____ / ____ \ / ____ / ____ \ / ____ / ____ \ / ____ / ____ \ / 3 \ / 3 \
| x / 11 \/ 17 | 4*x*\/ 26 | / 11 \/ 17 | 4*x*\/ 26 | / 88121 113*\/ 17 | 34424*x*\/ 26 | / 88121 113*\/ 17 | 34424*x*\/ 26 | / 4763 \/ 17 | 1868*x*\/ 26 | / 4763 \/ 17 | 1868*x*\/ 26 | 13*\22*x + 69*x/ 11*\4*x + 11*x/
| -------------------- dx = C - / ---- - ------ *atan|---------------------------------------------| - / ---- + ------ *atan|-----------------------------------------------| - 13* / --------- - ---------- *atan|-------------------------------------------------------------------| - 13* / --------- + ---------- *atan|-----------------------------------------------------------------| + 11* / ------- - ------ *atan|-----------------------------------------------------------| + 11* / ------- + ------ *atan|-------------------------------------------------------------| - --------------------------- + ------------------------
| 2 \/ 1768 1768 | _____________ _____________| \/ 1768 1768 | _____________ _____________| \/ 345403552 20317856 | _____________________ _____________________| \/ 345403552 20317856 | _____________________ _____________________| \/ 2043808 120224 | __________________ __________________| \/ 2043808 120224 | __________________ __________________| / 4 2\ / 4 2\
| / 4 2 \ | / ____ ____ / ____ | | / ____ ____ / ____ | | / ____ ____ / ____ | | / ____ ____ / ____ | | / ____ ____ / ____ | | / ____ ____ / ____ | 2*\5746 + 884*x + 4862*x / 2*\442 + 68*x + 374*x /
| \2*x + 11*x + 13/ \11*\/ 11 - \/ 17 + \/ 17 *\/ 11 - \/ 17 / \- 11*\/ 11 + \/ 17 + \/ 17 *\/ 11 + \/ 17 / \- 957*\/ 88121 - 1921*\/ 17 + 35*\/ 17 *\/ 88121 - 1921*\/ 17 / \957*\/ 88121 + 1921*\/ 17 + 35*\/ 17 *\/ 88121 + 1921*\/ 17 / \225*\/ 4763 - 17*\/ 17 + 11*\/ 17 *\/ 4763 - 17*\/ 17 / \- 225*\/ 4763 + 17*\/ 17 + 11*\/ 17 *\/ 4763 + 17*\/ 17 /
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/
∫ x 4 ( ( 2 x 4 + 11 x 2 ) + 13 ) 2 d x = C + 11 ( 4 x 3 + 11 x ) 2 ( 68 x 4 + 374 x 2 + 442 ) − 13 ( 22 x 3 + 69 x ) 2 ( 884 x 4 + 4862 x 2 + 5746 ) − 11 1768 − 17 1768 atan ( 4 26 x 17 11 − 17 + 11 11 − 17 ) + 11 4763 2043808 − 17 120224 atan ( 1868 26 x 11 17 4763 − 17 17 + 225 4763 − 17 17 ) − 13 113 17 20317856 + 88121 345403552 atan ( 34424 26 x 35 17 1921 17 + 88121 + 957 1921 17 + 88121 ) − 13 88121 345403552 − 113 17 20317856 atan ( 34424 26 x − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 ) − 17 1768 + 11 1768 atan ( 4 26 x − 11 17 + 11 + 17 17 + 11 ) + 11 17 120224 + 4763 2043808 atan ( 1868 26 x − 225 17 17 + 4763 + 11 17 17 17 + 4763 ) \int \frac{x^{4}}{\left(\left(2 x^{4} + 11 x^{2}\right) + 13\right)^{2}}\, dx = C + \frac{11 \left(4 x^{3} + 11 x\right)}{2 \left(68 x^{4} + 374 x^{2} + 442\right)} - \frac{13 \left(22 x^{3} + 69 x\right)}{2 \left(884 x^{4} + 4862 x^{2} + 5746\right)} - \sqrt{\frac{11}{1768} - \frac{\sqrt{17}}{1768}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{\sqrt{17} \sqrt{11 - \sqrt{17}} + 11 \sqrt{11 - \sqrt{17}}} \right)} + 11 \sqrt{\frac{4763}{2043808} - \frac{\sqrt{17}}{120224}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{11 \sqrt{17} \sqrt{4763 - 17 \sqrt{17}} + 225 \sqrt{4763 - 17 \sqrt{17}}} \right)} - 13 \sqrt{\frac{113 \sqrt{17}}{20317856} + \frac{88121}{345403552}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{35 \sqrt{17} \sqrt{1921 \sqrt{17} + 88121} + 957 \sqrt{1921 \sqrt{17} + 88121}} \right)} - 13 \sqrt{\frac{88121}{345403552} - \frac{113 \sqrt{17}}{20317856}} \operatorname{atan}{\left(\frac{34424 \sqrt{26} x}{- 957 \sqrt{88121 - 1921 \sqrt{17}} + 35 \sqrt{17} \sqrt{88121 - 1921 \sqrt{17}}} \right)} - \sqrt{\frac{\sqrt{17}}{1768} + \frac{11}{1768}} \operatorname{atan}{\left(\frac{4 \sqrt{26} x}{- 11 \sqrt{\sqrt{17} + 11} + \sqrt{17} \sqrt{\sqrt{17} + 11}} \right)} + 11 \sqrt{\frac{\sqrt{17}}{120224} + \frac{4763}{2043808}} \operatorname{atan}{\left(\frac{1868 \sqrt{26} x}{- 225 \sqrt{17 \sqrt{17} + 4763} + 11 \sqrt{17} \sqrt{17 \sqrt{17} + 4763}} \right)} ∫ ( ( 2 x 4 + 11 x 2 ) + 13 ) 2 x 4 d x = C + 2 ( 68 x 4 + 374 x 2 + 442 ) 11 ( 4 x 3 + 11 x ) − 2 ( 884 x 4 + 4862 x 2 + 5746 ) 13 ( 22 x 3 + 69 x ) − 1768 11 − 1768 17 atan ( 17 11 − 17 + 11 11 − 17 4 26 x ) + 11 2043808 4763 − 120224 17 atan ( 11 17 4763 − 17 17 + 225 4763 − 17 17 1868 26 x ) − 13 20317856 113 17 + 345403552 88121 atan ( 35 17 1921 17 + 88121 + 957 1921 17 + 88121 34424 26 x ) − 13 345403552 88121 − 20317856 113 17 atan ( − 957 88121 − 1921 17 + 35 17 88121 − 1921 17 34424 26 x ) − 1768 17 + 1768 11 atan ( − 11 17 + 11 + 17 17 + 11 4 26 x ) + 11 120224 17 + 2043808 4763 atan ( − 225 17 17 + 4763 + 11 17 17 17 + 4763 1868 26 x )
Gráfica
0.00 1.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.000 0.002
_________________ _________________
/ ____ / ____
37 / 4763 \/ 17 / 934 \ / 4763 \/ 17 / 934 \
--- + 2* / ------ - ------ *atan|---------------------------------------------------------| + 2* / ------ + ------ *atan|-------------------------------------------------------|
884 \/ 314432 18496 | __________________ __________________| \/ 314432 18496 | __________________ __________________|
| / ____ ____ / ____ | | / ____ ____ / ____ |
\- 22*\/ 4763 - 17*\/ 17 + \/ 17 *\/ 4763 - 17*\/ 17 / \22*\/ 4763 + 17*\/ 17 + \/ 17 *\/ 4763 + 17*\/ 17 /
2 4763 314432 − 17 18496 atan ( 934 − 22 4763 − 17 17 + 17 4763 − 17 17 ) + 37 884 + 2 17 18496 + 4763 314432 atan ( 934 17 17 17 + 4763 + 22 17 17 + 4763 ) 2 \sqrt{\frac{4763}{314432} - \frac{\sqrt{17}}{18496}} \operatorname{atan}{\left(\frac{934}{- 22 \sqrt{4763 - 17 \sqrt{17}} + \sqrt{17} \sqrt{4763 - 17 \sqrt{17}}} \right)} + \frac{37}{884} + 2 \sqrt{\frac{\sqrt{17}}{18496} + \frac{4763}{314432}} \operatorname{atan}{\left(\frac{934}{\sqrt{17} \sqrt{17 \sqrt{17} + 4763} + 22 \sqrt{17 \sqrt{17} + 4763}} \right)} 2 314432 4763 − 18496 17 atan ( − 22 4763 − 17 17 + 17 4763 − 17 17 934 ) + 884 37 + 2 18496 17 + 314432 4763 atan ( 17 17 17 + 4763 + 22 17 17 + 4763 934 )
=
_________________ _________________
/ ____ / ____
37 / 4763 \/ 17 / 934 \ / 4763 \/ 17 / 934 \
--- + 2* / ------ - ------ *atan|---------------------------------------------------------| + 2* / ------ + ------ *atan|-------------------------------------------------------|
884 \/ 314432 18496 | __________________ __________________| \/ 314432 18496 | __________________ __________________|
| / ____ ____ / ____ | | / ____ ____ / ____ |
\- 22*\/ 4763 - 17*\/ 17 + \/ 17 *\/ 4763 - 17*\/ 17 / \22*\/ 4763 + 17*\/ 17 + \/ 17 *\/ 4763 + 17*\/ 17 /
2 4763 314432 − 17 18496 atan ( 934 − 22 4763 − 17 17 + 17 4763 − 17 17 ) + 37 884 + 2 17 18496 + 4763 314432 atan ( 934 17 17 17 + 4763 + 22 17 17 + 4763 ) 2 \sqrt{\frac{4763}{314432} - \frac{\sqrt{17}}{18496}} \operatorname{atan}{\left(\frac{934}{- 22 \sqrt{4763 - 17 \sqrt{17}} + \sqrt{17} \sqrt{4763 - 17 \sqrt{17}}} \right)} + \frac{37}{884} + 2 \sqrt{\frac{\sqrt{17}}{18496} + \frac{4763}{314432}} \operatorname{atan}{\left(\frac{934}{\sqrt{17} \sqrt{17 \sqrt{17} + 4763} + 22 \sqrt{17 \sqrt{17} + 4763}} \right)} 2 314432 4763 − 18496 17 atan ( − 22 4763 − 17 17 + 17 4763 − 17 17 934 ) + 884 37 + 2 18496 17 + 314432 4763 atan ( 17 17 17 + 4763 + 22 17 17 + 4763 934 )
37/884 + 2*sqrt(4763/314432 - sqrt(17)/18496)*atan(934/(-22*sqrt(4763 - 17*sqrt(17)) + sqrt(17)*sqrt(4763 - 17*sqrt(17)))) + 2*sqrt(4763/314432 + sqrt(17)/18496)*atan(934/(22*sqrt(4763 + 17*sqrt(17)) + sqrt(17)*sqrt(4763 + 17*sqrt(17))))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.