Integral de 1/cosx^2(x+3) dx
Solución
Solución detallada
Vuelva a escribir el integrando:
x + 3 cos 2 ( x ) = x cos 2 ( x ) + 3 cos 2 ( x ) \frac{x + 3}{\cos^{2}{\left(x \right)}} = \frac{x}{\cos^{2}{\left(x \right)}} + \frac{3}{\cos^{2}{\left(x \right)}} c o s 2 ( x ) x + 3 = c o s 2 ( x ) x + c o s 2 ( x ) 3
Integramos término a término:
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 )
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ 3 cos 2 ( x ) d x = 3 ∫ 1 cos 2 ( x ) d x \int \frac{3}{\cos^{2}{\left(x \right)}}\, dx = 3 \int \frac{1}{\cos^{2}{\left(x \right)}}\, dx ∫ c o s 2 ( x ) 3 d x = 3 ∫ 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: 3 sin ( x ) cos ( x ) \frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)}} c o s ( x ) 3 s i n ( x )
El resultado es: − 2 x tan ( x 2 ) tan 2 ( x 2 ) − 1 + 3 sin ( x ) cos ( x ) + 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{3 \sin{\left(x \right)}}{\cos{\left(x \right)}} + \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 ) + c o s ( x ) 3 s i n ( 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 )
Ahora simplificar:
( 2 x + 6 ) ( tan 2 ( x 2 ) − 1 ) sin ( x ) + ( log ( tan ( x 2 ) − 1 ) + log ( tan ( x 2 ) + 1 ) ) ( cos ( x ) − 1 ) ( tan 2 ( x 2 ) − 1 ) + 2 ( − 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(2 x + 6\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin{\left(x \right)} + \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) + 2 \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 ) ( 2 x + 6 ) ( t a n 2 ( 2 x ) − 1 ) s i n ( x ) + ( 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 ) + 2 ( − 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:
( 2 x + 6 ) ( tan 2 ( x 2 ) − 1 ) sin ( x ) + ( log ( tan ( x 2 ) − 1 ) + log ( tan ( x 2 ) + 1 ) ) ( cos ( x ) − 1 ) ( tan 2 ( x 2 ) − 1 ) + 2 ( − 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(2 x + 6\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin{\left(x \right)} + \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) + 2 \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 ) ( 2 x + 6 ) ( t a n 2 ( 2 x ) − 1 ) s i n ( x ) + ( 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 ) + 2 ( − 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:
( 2 x + 6 ) ( tan 2 ( x 2 ) − 1 ) sin ( x ) + ( log ( tan ( x 2 ) − 1 ) + log ( tan ( x 2 ) + 1 ) ) ( cos ( x ) − 1 ) ( tan 2 ( x 2 ) − 1 ) + 2 ( − 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(2 x + 6\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin{\left(x \right)} + \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) + 2 \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 ) ( 2 x + 6 ) ( t a n 2 ( 2 x ) − 1 ) s i n ( x ) + ( 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 ) + 2 ( − 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]
/ / 2/x\\ / /x\\ / /x\\ 2/x\ / /x\\ 2/x\ / /x\\ 2/x\ / 2/x\\ /x\
| log|1 + tan |-|| log|1 + tan|-|| log|-1 + tan|-|| tan |-|*log|1 + tan|-|| tan |-|*log|-1 + tan|-|| tan |-|*log|1 + tan |-|| 2*x*tan|-|
| x + 3 \ \2// \ \2// \ \2// 3*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/
/
∫ x + 3 cos 2 ( x ) d x = C − 2 x tan ( x 2 ) tan 2 ( x 2 ) − 1 + 3 sin ( x ) cos ( x ) + 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 \int \frac{x + 3}{\cos^{2}{\left(x \right)}}\, dx = C - \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)}} + \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} ∫ cos 2 ( x ) x + 3 d x = C − tan 2 ( 2 x ) − 1 2 x tan ( 2 x ) + cos ( x ) 3 sin ( x ) + tan 2 ( 2 x ) − 1 log ( tan ( 2 x ) − 1 ) tan 2 ( 2 x ) − tan 2 ( 2 x ) − 1 log ( tan ( 2 x ) − 1 ) + tan 2 ( 2 x ) − 1 log ( tan ( 2 x ) + 1 ) tan 2 ( 2 x ) − tan 2 ( 2 x ) − 1 log ( tan ( 2 x ) + 1 ) − tan 2 ( 2 x ) − 1 log ( tan 2 ( 2 x ) + 1 ) tan 2 ( 2 x ) + tan 2 ( 2 x ) − 1 log ( tan 2 ( 2 x ) + 1 )
Gráfica
0.00 1.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0 20
/ 2 \ 2 2 2 / 2 \
log\1 + tan (1/2)/ pi*I + log(1 - tan(1/2)) log(1 + tan(1/2)) 8*tan(1/2) tan (1/2)*(pi*I + log(1 - tan(1/2))) tan (1/2)*log(1 + tan(1/2)) tan (1/2)*log\1 + tan (1/2)/
------------------ - pi*I - ------------------------ - ----------------- - -------------- + ------------------------------------ + --------------------------- - ----------------------------
2 2 2 2 2 2 2
-1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2)
log ( tan 2 ( 1 2 ) + 1 ) − 1 + tan 2 ( 1 2 ) + log ( tan ( 1 2 ) + 1 ) tan 2 ( 1 2 ) − 1 + tan 2 ( 1 2 ) − log ( tan 2 ( 1 2 ) + 1 ) tan 2 ( 1 2 ) − 1 + tan 2 ( 1 2 ) − log ( tan ( 1 2 ) + 1 ) − 1 + tan 2 ( 1 2 ) − 8 tan ( 1 2 ) − 1 + tan 2 ( 1 2 ) − i π + ( log ( 1 − tan ( 1 2 ) ) + i π ) tan 2 ( 1 2 ) − 1 + tan 2 ( 1 2 ) − log ( 1 − tan ( 1 2 ) ) + i π − 1 + tan 2 ( 1 2 ) \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{8 \tan{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - i \pi + \frac{\left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} − 1 + tan 2 ( 2 1 ) log ( tan 2 ( 2 1 ) + 1 ) + − 1 + tan 2 ( 2 1 ) log ( tan ( 2 1 ) + 1 ) tan 2 ( 2 1 ) − − 1 + tan 2 ( 2 1 ) log ( tan 2 ( 2 1 ) + 1 ) tan 2 ( 2 1 ) − − 1 + tan 2 ( 2 1 ) log ( tan ( 2 1 ) + 1 ) − − 1 + tan 2 ( 2 1 ) 8 tan ( 2 1 ) − iπ + − 1 + tan 2 ( 2 1 ) ( log ( 1 − tan ( 2 1 ) ) + iπ ) tan 2 ( 2 1 ) − − 1 + tan 2 ( 2 1 ) log ( 1 − tan ( 2 1 ) ) + iπ
=
/ 2 \ 2 2 2 / 2 \
log\1 + tan (1/2)/ pi*I + log(1 - tan(1/2)) log(1 + tan(1/2)) 8*tan(1/2) tan (1/2)*(pi*I + log(1 - tan(1/2))) tan (1/2)*log(1 + tan(1/2)) tan (1/2)*log\1 + tan (1/2)/
------------------ - pi*I - ------------------------ - ----------------- - -------------- + ------------------------------------ + --------------------------- - ----------------------------
2 2 2 2 2 2 2
-1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2) -1 + tan (1/2)
log ( tan 2 ( 1 2 ) + 1 ) − 1 + tan 2 ( 1 2 ) + log ( tan ( 1 2 ) + 1 ) tan 2 ( 1 2 ) − 1 + tan 2 ( 1 2 ) − log ( tan 2 ( 1 2 ) + 1 ) tan 2 ( 1 2 ) − 1 + tan 2 ( 1 2 ) − log ( tan ( 1 2 ) + 1 ) − 1 + tan 2 ( 1 2 ) − 8 tan ( 1 2 ) − 1 + tan 2 ( 1 2 ) − i π + ( log ( 1 − tan ( 1 2 ) ) + i π ) tan 2 ( 1 2 ) − 1 + tan 2 ( 1 2 ) − log ( 1 − tan ( 1 2 ) ) + i π − 1 + tan 2 ( 1 2 ) \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{8 \tan{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - i \pi + \frac{\left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} − 1 + tan 2 ( 2 1 ) log ( tan 2 ( 2 1 ) + 1 ) + − 1 + tan 2 ( 2 1 ) log ( tan ( 2 1 ) + 1 ) tan 2 ( 2 1 ) − − 1 + tan 2 ( 2 1 ) log ( tan 2 ( 2 1 ) + 1 ) tan 2 ( 2 1 ) − − 1 + tan 2 ( 2 1 ) log ( tan ( 2 1 ) + 1 ) − − 1 + tan 2 ( 2 1 ) 8 tan ( 2 1 ) − iπ + − 1 + tan 2 ( 2 1 ) ( log ( 1 − tan ( 2 1 ) ) + iπ ) tan 2 ( 2 1 ) − − 1 + tan 2 ( 2 1 ) log ( 1 − tan ( 2 1 ) ) + iπ
log(1 + tan(1/2)^2)/(-1 + tan(1/2)^2) - pi*i - (pi*i + log(1 - tan(1/2)))/(-1 + tan(1/2)^2) - log(1 + tan(1/2))/(-1 + tan(1/2)^2) - 8*tan(1/2)/(-1 + tan(1/2)^2) + tan(1/2)^2*(pi*i + log(1 - tan(1/2)))/(-1 + tan(1/2)^2) + tan(1/2)^2*log(1 + tan(1/2))/(-1 + tan(1/2)^2) - tan(1/2)^2*log(1 + tan(1/2)^2)/(-1 + tan(1/2)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.
