Integral de (3x+5)/(cosx)^2 dx
Solución
Solución detallada
Vuelva a escribir el integrando:
3 x + 5 cos 2 ( x ) = 3 x cos 2 ( x ) + 5 cos 2 ( x ) \frac{3 x + 5}{\cos^{2}{\left(x \right)}} = \frac{3 x}{\cos^{2}{\left(x \right)}} + \frac{5}{\cos^{2}{\left(x \right)}} c o s 2 ( x ) 3 x + 5 = c o s 2 ( x ) 3 x + c o s 2 ( x ) 5
Integramos término a término:
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ 3 x cos 2 ( x ) d x = 3 ∫ x cos 2 ( x ) d x \int \frac{3 x}{\cos^{2}{\left(x \right)}}\, dx = 3 \int \frac{x}{\cos^{2}{\left(x \right)}}\, dx ∫ c o s 2 ( x ) 3 x d x = 3 ∫ c o s 2 ( x ) x d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
− 2 x tan ( x 2 ) tan 2 ( x 2 ) − 1 + log ( tan ( x 2 ) − 1 ) tan 2 ( x 2 ) tan 2 ( x 2 ) − 1 − log ( tan ( x 2 ) − 1 ) tan 2 ( x 2 ) − 1 + log ( tan ( x 2 ) + 1 ) tan 2 ( x 2 ) tan 2 ( x 2 ) − 1 − log ( tan ( x 2 ) + 1 ) tan 2 ( x 2 ) − 1 − log ( tan 2 ( x 2 ) + 1 ) tan 2 ( x 2 ) tan 2 ( x 2 ) − 1 + log ( tan 2 ( x 2 ) + 1 ) tan 2 ( x 2 ) − 1 - \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} − t a n 2 ( 2 x ) − 1 2 x t a n ( 2 x ) + t a n 2 ( 2 x ) − 1 l o g ( t a n ( 2 x ) − 1 ) t a n 2 ( 2 x ) − t a n 2 ( 2 x ) − 1 l o g ( t a n ( 2 x ) − 1 ) + t a n 2 ( 2 x ) − 1 l o g ( t a n ( 2 x ) + 1 ) t a n 2 ( 2 x ) − t a n 2 ( 2 x ) − 1 l o g ( t a n ( 2 x ) + 1 ) − t a n 2 ( 2 x ) − 1 l o g ( t a n 2 ( 2 x ) + 1 ) t a n 2 ( 2 x ) + t a n 2 ( 2 x ) − 1 l o g ( t a n 2 ( 2 x ) + 1 )
Por lo tanto, el resultado es: − 6 x tan ( x 2 ) tan 2 ( x 2 ) − 1 + 3 log ( tan ( x 2 ) − 1 ) tan 2 ( x 2 ) tan 2 ( x 2 ) − 1 − 3 log ( tan ( x 2 ) − 1 ) tan 2 ( x 2 ) − 1 + 3 log ( tan ( x 2 ) + 1 ) tan 2 ( x 2 ) tan 2 ( x 2 ) − 1 − 3 log ( tan ( x 2 ) + 1 ) tan 2 ( x 2 ) − 1 − 3 log ( tan 2 ( x 2 ) + 1 ) tan 2 ( x 2 ) tan 2 ( x 2 ) − 1 + 3 log ( tan 2 ( x 2 ) + 1 ) tan 2 ( x 2 ) − 1 - \frac{6 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{3 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{3 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{3 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{3 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{3 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{3 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} − t a n 2 ( 2 x ) − 1 6 x t a n ( 2 x ) + t a n 2 ( 2 x ) − 1 3 l o g ( t a n ( 2 x ) − 1 ) t a n 2 ( 2 x ) − t a n 2 ( 2 x ) − 1 3 l o g ( t a n ( 2 x ) − 1 ) + t a n 2 ( 2 x ) − 1 3 l o g ( t a n ( 2 x ) + 1 ) t a n 2 ( 2 x ) − t a n 2 ( 2 x ) − 1 3 l o g ( t a n ( 2 x ) + 1 ) − t a n 2 ( 2 x ) − 1 3 l o g ( t a n 2 ( 2 x ) + 1 ) t a n 2 ( 2 x ) + t a n 2 ( 2 x ) − 1 3 l o g ( t a n 2 ( 2 x ) + 1 )
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ 5 cos 2 ( x ) d x = 5 ∫ 1 cos 2 ( x ) d x \int \frac{5}{\cos^{2}{\left(x \right)}}\, dx = 5 \int \frac{1}{\cos^{2}{\left(x \right)}}\, dx ∫ c o s 2 ( x ) 5 d x = 5 ∫ c o s 2 ( x ) 1 d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
sin ( x ) cos ( x ) \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} c o s ( x ) s i n ( x )
Por lo tanto, el resultado es: 5 sin ( x ) cos ( x ) \frac{5 \sin{\left(x \right)}}{\cos{\left(x \right)}} c o s ( x ) 5 s i n ( x )
El resultado es: − 6 x tan ( x 2 ) tan 2 ( x 2 ) − 1 + 5 sin ( x ) cos ( x ) + 3 log ( tan ( x 2 ) − 1 ) tan 2 ( x 2 ) tan 2 ( x 2 ) − 1 − 3 log ( tan ( x 2 ) − 1 ) tan 2 ( x 2 ) − 1 + 3 log ( tan ( x 2 ) + 1 ) tan 2 ( x 2 ) tan 2 ( x 2 ) − 1 − 3 log ( tan ( x 2 ) + 1 ) tan 2 ( x 2 ) − 1 − 3 log ( tan 2 ( x 2 ) + 1 ) tan 2 ( x 2 ) tan 2 ( x 2 ) − 1 + 3 log ( tan 2 ( x 2 ) + 1 ) tan 2 ( x 2 ) − 1 - \frac{6 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{5 \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{3 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{3 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{3 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{3 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{3 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{3 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} − t a n 2 ( 2 x ) − 1 6 x t a n ( 2 x ) + c o s ( x ) 5 s i n ( x ) + t a n 2 ( 2 x ) − 1 3 l o g ( t a n ( 2 x ) − 1 ) t a n 2 ( 2 x ) − t a n 2 ( 2 x ) − 1 3 l o g ( t a n ( 2 x ) − 1 ) + t a n 2 ( 2 x ) − 1 3 l o g ( t a n ( 2 x ) + 1 ) t a n 2 ( 2 x ) − t a n 2 ( 2 x ) − 1 3 l o g ( t a n ( 2 x ) + 1 ) − t a n 2 ( 2 x ) − 1 3 l o g ( t a n 2 ( 2 x ) + 1 ) t a n 2 ( 2 x ) + t a n 2 ( 2 x ) − 1 3 l o g ( t a n 2 ( 2 x ) + 1 )
Ahora simplificar:
( 6 x + 10 ) ( tan 2 ( x 2 ) − 1 ) sin ( x ) + 3 ( log ( tan ( x 2 ) − 1 ) + log ( tan ( x 2 ) + 1 ) ) ( cos ( x ) − 1 ) ( tan 2 ( x 2 ) − 1 ) + 6 ( − log ( 2 cos ( x ) + 1 ) tan 2 ( x 2 ) + log ( 2 cos ( x ) + 1 ) − log ( tan ( x 2 ) − 1 ) − log ( tan ( x 2 ) + 1 ) ) cos ( x ) 2 ( tan 2 ( x 2 ) − 1 ) cos ( x ) \frac{\left(6 x + 10\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin{\left(x \right)} + 3 \left(\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \left(\cos{\left(x \right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) + 6 \left(- \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \cos{\left(x \right)}}{2 \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \cos{\left(x \right)}} 2 ( t a n 2 ( 2 x ) − 1 ) c o s ( x ) ( 6 x + 10 ) ( t a n 2 ( 2 x ) − 1 ) s i n ( x ) + 3 ( l o g ( t a n ( 2 x ) − 1 ) + l o g ( t a n ( 2 x ) + 1 ) ) ( c o s ( x ) − 1 ) ( t a n 2 ( 2 x ) − 1 ) + 6 ( − l o g ( c o s ( x ) + 1 2 ) t a n 2 ( 2 x ) + l o g ( c o s ( x ) + 1 2 ) − l o g ( t a n ( 2 x ) − 1 ) − l o g ( t a n ( 2 x ) + 1 ) ) c o s ( x )
Añadimos la constante de integración:
( 6 x + 10 ) ( tan 2 ( x 2 ) − 1 ) sin ( x ) + 3 ( log ( tan ( x 2 ) − 1 ) + log ( tan ( x 2 ) + 1 ) ) ( cos ( x ) − 1 ) ( tan 2 ( x 2 ) − 1 ) + 6 ( − log ( 2 cos ( x ) + 1 ) tan 2 ( x 2 ) + log ( 2 cos ( x ) + 1 ) − log ( tan ( x 2 ) − 1 ) − log ( tan ( x 2 ) + 1 ) ) cos ( x ) 2 ( tan 2 ( x 2 ) − 1 ) cos ( x ) + c o n s t a n t \frac{\left(6 x + 10\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin{\left(x \right)} + 3 \left(\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \left(\cos{\left(x \right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) + 6 \left(- \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \cos{\left(x \right)}}{2 \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \cos{\left(x \right)}}+ \mathrm{constant} 2 ( t a n 2 ( 2 x ) − 1 ) c o s ( x ) ( 6 x + 10 ) ( t a n 2 ( 2 x ) − 1 ) s i n ( x ) + 3 ( l o g ( t a n ( 2 x ) − 1 ) + l o g ( t a n ( 2 x ) + 1 ) ) ( c o s ( x ) − 1 ) ( t a n 2 ( 2 x ) − 1 ) + 6 ( − l o g ( c o s ( x ) + 1 2 ) t a n 2 ( 2 x ) + l o g ( c o s ( x ) + 1 2 ) − l o g ( t a n ( 2 x ) − 1 ) − l o g ( t a n ( 2 x ) + 1 ) ) c o s ( x ) + constant
Respuesta:
( 6 x + 10 ) ( tan 2 ( x 2 ) − 1 ) sin ( x ) + 3 ( log ( tan ( x 2 ) − 1 ) + log ( tan ( x 2 ) + 1 ) ) ( cos ( x ) − 1 ) ( tan 2 ( x 2 ) − 1 ) + 6 ( − log ( 2 cos ( x ) + 1 ) tan 2 ( x 2 ) + log ( 2 cos ( x ) + 1 ) − log ( tan ( x 2 ) − 1 ) − log ( tan ( x 2 ) + 1 ) ) cos ( x ) 2 ( tan 2 ( x 2 ) − 1 ) cos ( x ) + c o n s t a n t \frac{\left(6 x + 10\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin{\left(x \right)} + 3 \left(\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \left(\cos{\left(x \right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) + 6 \left(- \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \cos{\left(x \right)}}{2 \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \cos{\left(x \right)}}+ \mathrm{constant} 2 ( t a n 2 ( 2 x ) − 1 ) c o s ( x ) ( 6 x + 10 ) ( t a n 2 ( 2 x ) − 1 ) s i n ( x ) + 3 ( l o g ( t a n ( 2 x ) − 1 ) + l o g ( t a n ( 2 x ) + 1 ) ) ( c o s ( x ) − 1 ) ( t a n 2 ( 2 x ) − 1 ) + 6 ( − l o g ( c o s ( x ) + 1 2 ) t a n 2 ( 2 x ) + l o g ( c o s ( x ) + 1 2 ) − l o g ( t a n ( 2 x ) − 1 ) − l o g ( t a n ( 2 x ) + 1 ) ) c o s ( x ) + constant
Respuesta (Indefinida)
[src]
/ / /x\\ / /x\\ / 2/x\\ /x\ 2/x\ / 2/x\\ 2/x\ / /x\\ 2/x\ / /x\\
| 3*log|1 + tan|-|| 3*log|-1 + tan|-|| 3*log|1 + tan |-|| 6*x*tan|-| 3*tan |-|*log|1 + tan |-|| 3*tan |-|*log|1 + tan|-|| 3*tan |-|*log|-1 + tan|-||
| 3*x + 5 \ \2// \ \2// \ \2// 5*sin(x) \2/ \2/ \ \2// \2/ \ \2// \2/ \ \2//
| ------- dx = C - ----------------- - ------------------ + ------------------ + -------- - ------------ - -------------------------- + ------------------------- + --------------------------
| 2 2/x\ 2/x\ 2/x\ cos(x) 2/x\ 2/x\ 2/x\ 2/x\
| cos (x) -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-| -1 + tan |-|
| \2/ \2/ \2/ \2/ \2/ \2/ \2/
/
∫ 3 x + 5 cos 2 ( x ) d x = C − 6 x tan ( x 2 ) tan 2 ( x 2 ) − 1 + 5 sin ( x ) cos ( x ) + 3 log ( tan ( x 2 ) − 1 ) tan 2 ( x 2 ) tan 2 ( x 2 ) − 1 − 3 log ( tan ( x 2 ) − 1 ) tan 2 ( x 2 ) − 1 + 3 log ( tan ( x 2 ) + 1 ) tan 2 ( x 2 ) tan 2 ( x 2 ) − 1 − 3 log ( tan ( x 2 ) + 1 ) tan 2 ( x 2 ) − 1 − 3 log ( tan 2 ( x 2 ) + 1 ) tan 2 ( x 2 ) tan 2 ( x 2 ) − 1 + 3 log ( tan 2 ( x 2 ) + 1 ) tan 2 ( x 2 ) − 1 \int \frac{3 x + 5}{\cos^{2}{\left(x \right)}}\, dx = C - \frac{6 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{5 \sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{3 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{3 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{3 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{3 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{3 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{3 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} ∫ cos 2 ( x ) 3 x + 5 d x = C − tan 2 ( 2 x ) − 1 6 x tan ( 2 x ) + cos ( x ) 5 sin ( x ) + tan 2 ( 2 x ) − 1 3 log ( tan ( 2 x ) − 1 ) tan 2 ( 2 x ) − tan 2 ( 2 x ) − 1 3 log ( tan ( 2 x ) − 1 ) + tan 2 ( 2 x ) − 1 3 log ( tan ( 2 x ) + 1 ) tan 2 ( 2 x ) − tan 2 ( 2 x ) − 1 3 log ( tan ( 2 x ) + 1 ) − tan 2 ( 2 x ) − 1 3 log ( tan 2 ( 2 x ) + 1 ) tan 2 ( 2 x ) + tan 2 ( 2 x ) − 1 3 log ( tan 2 ( 2 x ) + 1 )
Gráfica
0.7860 0.7870 0.7880 0.7890 0.7900 0.7910 0.7920 0.7930 0.7940 0.7950 14.71 14.73
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.
