Integral de 1/2*sqrt(x)+1/(sqrt(x)+x)^3 dx
Solución
Solución detallada
Integramos término a término:
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ x 2 d x = ∫ x d x 2 \int \frac{\sqrt{x}}{2}\, dx = \frac{\int \sqrt{x}\, dx}{2} ∫ 2 x d x = 2 ∫ x d x
Integral x n x^{n} x n x n + 1 n + 1 \frac{x^{n + 1}}{n + 1} n + 1 x n + 1 n ≠ − 1 n \neq -1 n = − 1
∫ x d x = 2 x 3 2 3 \int \sqrt{x}\, dx = \frac{2 x^{\frac{3}{2}}}{3} ∫ x d x = 3 2 x 2 3
Por lo tanto, el resultado es: x 3 2 3 \frac{x^{\frac{3}{2}}}{3} 3 x 2 3
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
− 6 x 3 2 log ( x ) 2 x 3 2 + x 2 + x + 12 x 3 2 log ( x + 1 ) 2 x 3 2 + x 2 + x − 6 x 3 2 2 x 3 2 + x 2 + x − 2 x 2 x 3 2 + x 2 + x − 3 x 2 log ( x ) 2 x 3 2 + x 2 + x + 6 x 2 log ( x + 1 ) 2 x 3 2 + x 2 + x − 3 x log ( x ) 2 x 3 2 + x 2 + x + 6 x log ( x + 1 ) 2 x 3 2 + x 2 + x − 9 x 2 x 3 2 + x 2 + x - \frac{6 x^{\frac{3}{2}} \log{\left(x \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} + \frac{12 x^{\frac{3}{2}} \log{\left(\sqrt{x} + 1 \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{6 x^{\frac{3}{2}}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{2 \sqrt{x}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{3 x^{2} \log{\left(x \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} + \frac{6 x^{2} \log{\left(\sqrt{x} + 1 \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{3 x \log{\left(x \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} + \frac{6 x \log{\left(\sqrt{x} + 1 \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{9 x}{2 x^{\frac{3}{2}} + x^{2} + x} − 2 x 2 3 + x 2 + x 6 x 2 3 l o g ( x ) + 2 x 2 3 + x 2 + x 12 x 2 3 l o g ( x + 1 ) − 2 x 2 3 + x 2 + x 6 x 2 3 − 2 x 2 3 + x 2 + x 2 x − 2 x 2 3 + x 2 + x 3 x 2 l o g ( x ) + 2 x 2 3 + x 2 + x 6 x 2 l o g ( x + 1 ) − 2 x 2 3 + x 2 + x 3 x l o g ( x ) + 2 x 2 3 + x 2 + x 6 x l o g ( x + 1 ) − 2 x 2 3 + x 2 + x 9 x
El resultado es: x 3 2 3 − 6 x 3 2 log ( x ) 2 x 3 2 + x 2 + x + 12 x 3 2 log ( x + 1 ) 2 x 3 2 + x 2 + x − 6 x 3 2 2 x 3 2 + x 2 + x − 2 x 2 x 3 2 + x 2 + x − 3 x 2 log ( x ) 2 x 3 2 + x 2 + x + 6 x 2 log ( x + 1 ) 2 x 3 2 + x 2 + x − 3 x log ( x ) 2 x 3 2 + x 2 + x + 6 x log ( x + 1 ) 2 x 3 2 + x 2 + x − 9 x 2 x 3 2 + x 2 + x \frac{x^{\frac{3}{2}}}{3} - \frac{6 x^{\frac{3}{2}} \log{\left(x \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} + \frac{12 x^{\frac{3}{2}} \log{\left(\sqrt{x} + 1 \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{6 x^{\frac{3}{2}}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{2 \sqrt{x}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{3 x^{2} \log{\left(x \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} + \frac{6 x^{2} \log{\left(\sqrt{x} + 1 \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{3 x \log{\left(x \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} + \frac{6 x \log{\left(\sqrt{x} + 1 \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{9 x}{2 x^{\frac{3}{2}} + x^{2} + x} 3 x 2 3 − 2 x 2 3 + x 2 + x 6 x 2 3 l o g ( x ) + 2 x 2 3 + x 2 + x 12 x 2 3 l o g ( x + 1 ) − 2 x 2 3 + x 2 + x 6 x 2 3 − 2 x 2 3 + x 2 + x 2 x − 2 x 2 3 + x 2 + x 3 x 2 l o g ( x ) + 2 x 2 3 + x 2 + x 6 x 2 l o g ( x + 1 ) − 2 x 2 3 + x 2 + x 3 x l o g ( x ) + 2 x 2 3 + x 2 + x 6 x l o g ( x + 1 ) − 2 x 2 3 + x 2 + x 9 x
Ahora simplificar:
x 7 2 + x 5 2 − 18 x 3 2 log ( x ) + 36 x 3 2 log ( x + 1 ) − 18 x 3 2 − 6 x + 2 x 3 − 9 x 2 log ( x ) + 18 x 2 log ( x + 1 ) − 9 x log ( x ) + 18 x log ( x + 1 ) − 27 x 3 ( 2 x 3 2 + x 2 + x ) \frac{x^{\frac{7}{2}} + x^{\frac{5}{2}} - 18 x^{\frac{3}{2}} \log{\left(x \right)} + 36 x^{\frac{3}{2}} \log{\left(\sqrt{x} + 1 \right)} - 18 x^{\frac{3}{2}} - 6 \sqrt{x} + 2 x^{3} - 9 x^{2} \log{\left(x \right)} + 18 x^{2} \log{\left(\sqrt{x} + 1 \right)} - 9 x \log{\left(x \right)} + 18 x \log{\left(\sqrt{x} + 1 \right)} - 27 x}{3 \left(2 x^{\frac{3}{2}} + x^{2} + x\right)} 3 ( 2 x 2 3 + x 2 + x ) x 2 7 + x 2 5 − 18 x 2 3 l o g ( x ) + 36 x 2 3 l o g ( x + 1 ) − 18 x 2 3 − 6 x + 2 x 3 − 9 x 2 l o g ( x ) + 18 x 2 l o g ( x + 1 ) − 9 x l o g ( x ) + 18 x l o g ( x + 1 ) − 27 x
Añadimos la constante de integración:
x 7 2 + x 5 2 − 18 x 3 2 log ( x ) + 36 x 3 2 log ( x + 1 ) − 18 x 3 2 − 6 x + 2 x 3 − 9 x 2 log ( x ) + 18 x 2 log ( x + 1 ) − 9 x log ( x ) + 18 x log ( x + 1 ) − 27 x 3 ( 2 x 3 2 + x 2 + x ) + c o n s t a n t \frac{x^{\frac{7}{2}} + x^{\frac{5}{2}} - 18 x^{\frac{3}{2}} \log{\left(x \right)} + 36 x^{\frac{3}{2}} \log{\left(\sqrt{x} + 1 \right)} - 18 x^{\frac{3}{2}} - 6 \sqrt{x} + 2 x^{3} - 9 x^{2} \log{\left(x \right)} + 18 x^{2} \log{\left(\sqrt{x} + 1 \right)} - 9 x \log{\left(x \right)} + 18 x \log{\left(\sqrt{x} + 1 \right)} - 27 x}{3 \left(2 x^{\frac{3}{2}} + x^{2} + x\right)}+ \mathrm{constant} 3 ( 2 x 2 3 + x 2 + x ) x 2 7 + x 2 5 − 18 x 2 3 l o g ( x ) + 36 x 2 3 l o g ( x + 1 ) − 18 x 2 3 − 6 x + 2 x 3 − 9 x 2 l o g ( x ) + 18 x 2 l o g ( x + 1 ) − 9 x l o g ( x ) + 18 x l o g ( x + 1 ) − 27 x + constant
Respuesta:
x 7 2 + x 5 2 − 18 x 3 2 log ( x ) + 36 x 3 2 log ( x + 1 ) − 18 x 3 2 − 6 x + 2 x 3 − 9 x 2 log ( x ) + 18 x 2 log ( x + 1 ) − 9 x log ( x ) + 18 x log ( x + 1 ) − 27 x 3 ( 2 x 3 2 + x 2 + x ) + c o n s t a n t \frac{x^{\frac{7}{2}} + x^{\frac{5}{2}} - 18 x^{\frac{3}{2}} \log{\left(x \right)} + 36 x^{\frac{3}{2}} \log{\left(\sqrt{x} + 1 \right)} - 18 x^{\frac{3}{2}} - 6 \sqrt{x} + 2 x^{3} - 9 x^{2} \log{\left(x \right)} + 18 x^{2} \log{\left(\sqrt{x} + 1 \right)} - 9 x \log{\left(x \right)} + 18 x \log{\left(\sqrt{x} + 1 \right)} - 27 x}{3 \left(2 x^{\frac{3}{2}} + x^{2} + x\right)}+ \mathrm{constant} 3 ( 2 x 2 3 + x 2 + x ) x 2 7 + x 2 5 − 18 x 2 3 l o g ( x ) + 36 x 2 3 l o g ( x + 1 ) − 18 x 2 3 − 6 x + 2 x 3 − 9 x 2 l o g ( x ) + 18 x 2 l o g ( x + 1 ) − 9 x l o g ( x ) + 18 x l o g ( x + 1 ) − 27 x + constant
Respuesta (Indefinida)
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| / ___ \ 3/2 3/2 ___ 3/2 2 / ___\ 2 / ___\ 3/2 / ___\
| |\/ x 1 | x 9*x 6*x 2*\/ x 6*x *log(x) 3*x*log(x) 3*x *log(x) 6*x*log\1 + \/ x / 6*x *log\1 + \/ x / 12*x *log\1 + \/ x /
| |----- + ------------| dx = C + ---- - --------------- - --------------- - --------------- - --------------- - --------------- - --------------- + ------------------ + ------------------- + ----------------------
| | 2 3| 3 2 3/2 2 3/2 2 3/2 2 3/2 2 3/2 2 3/2 2 3/2 2 3/2 2 3/2
| | / ___ \ | x + x + 2*x x + x + 2*x x + x + 2*x x + x + 2*x x + x + 2*x x + x + 2*x x + x + 2*x x + x + 2*x x + x + 2*x
| \ \\/ x + x/ /
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∫ ( x 2 + 1 ( x + x ) 3 ) d x = C + x 3 2 3 − 6 x 3 2 log ( x ) 2 x 3 2 + x 2 + x + 12 x 3 2 log ( x + 1 ) 2 x 3 2 + x 2 + x − 6 x 3 2 2 x 3 2 + x 2 + x − 2 x 2 x 3 2 + x 2 + x − 3 x 2 log ( x ) 2 x 3 2 + x 2 + x + 6 x 2 log ( x + 1 ) 2 x 3 2 + x 2 + x − 3 x log ( x ) 2 x 3 2 + x 2 + x + 6 x log ( x + 1 ) 2 x 3 2 + x 2 + x − 9 x 2 x 3 2 + x 2 + x \int \left(\frac{\sqrt{x}}{2} + \frac{1}{\left(\sqrt{x} + x\right)^{3}}\right)\, dx = C + \frac{x^{\frac{3}{2}}}{3} - \frac{6 x^{\frac{3}{2}} \log{\left(x \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} + \frac{12 x^{\frac{3}{2}} \log{\left(\sqrt{x} + 1 \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{6 x^{\frac{3}{2}}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{2 \sqrt{x}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{3 x^{2} \log{\left(x \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} + \frac{6 x^{2} \log{\left(\sqrt{x} + 1 \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{3 x \log{\left(x \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} + \frac{6 x \log{\left(\sqrt{x} + 1 \right)}}{2 x^{\frac{3}{2}} + x^{2} + x} - \frac{9 x}{2 x^{\frac{3}{2}} + x^{2} + x} ∫ ( 2 x + ( x + x ) 3 1 ) d x = C + 3 x 2 3 − 2 x 2 3 + x 2 + x 6 x 2 3 log ( x ) + 2 x 2 3 + x 2 + x 12 x 2 3 log ( x + 1 ) − 2 x 2 3 + x 2 + x 6 x 2 3 − 2 x 2 3 + x 2 + x 2 x − 2 x 2 3 + x 2 + x 3 x 2 log ( x ) + 2 x 2 3 + x 2 + x 6 x 2 log ( x + 1 ) − 2 x 2 3 + x 2 + x 3 x log ( x ) + 2 x 2 3 + x 2 + x 6 x log ( x + 1 ) − 2 x 2 3 + x 2 + x 9 x
Gráfica
1.00 4.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 0.0 5.0
149
--- - 6*log(2) - 3*log(4) + 6*log(3)
36
− 6 log ( 2 ) − 3 log ( 4 ) + 149 36 + 6 log ( 3 ) - 6 \log{\left(2 \right)} - 3 \log{\left(4 \right)} + \frac{149}{36} + 6 \log{\left(3 \right)} − 6 log ( 2 ) − 3 log ( 4 ) + 36 149 + 6 log ( 3 )
=
149
--- - 6*log(2) - 3*log(4) + 6*log(3)
36
− 6 log ( 2 ) − 3 log ( 4 ) + 149 36 + 6 log ( 3 ) - 6 \log{\left(2 \right)} - 3 \log{\left(4 \right)} + \frac{149}{36} + 6 \log{\left(3 \right)} − 6 log ( 2 ) − 3 log ( 4 ) + 36 149 + 6 log ( 3 )
149/36 - 6*log(2) - 3*log(4) + 6*log(3)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.
