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Resolución de integrales/ (cosx)/ x/(cosx)^2

Integral de x/(cosx)^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi           
 --           
 4            
  /           
 |            
 |     x      
 |  ------- dx
 |     2      
 |  cos (x)   
 |            
/             
0
∫0π4xcos⁡2(x) dx\int\limits_{0}^{\frac{\pi}{4}} \frac{x}{\cos^{2}{\left(x \right)}}\, dx 
Integral(x/cos(x)^2, (x, 0, pi/4))
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                \        \2//      \       \2//      \        \2//       \2/    \       \2//       \2/    \        \2//       \2/    \        \2//           \2/ 
 | ------- dx = C + ---------------- - --------------- - ---------------- + ----------------------- + ------------------------ - ------------------------ - ------------
 |    2                       2/x\               2/x\              2/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/
/
∫xcos⁡2(x) dx=C−2xtan⁡(x2)tan⁡2(x2)−1+log⁡(tan⁡(x2)−1)tan⁡2(x2)tan⁡2(x2)−1−log⁡(tan⁡(x2)−1)tan⁡2(x2)−1+log⁡(tan⁡(x2)+1)tan⁡2(x2)tan⁡2(x2)−1−log⁡(tan⁡(x2)+1)tan⁡2(x2)−1−log⁡(tan⁡2(x2)+1)tan⁡2(x2)tan⁡2(x2)−1+log⁡(tan⁡2(x2)+1)tan⁡2(x2)−1\int \frac{x}{\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{\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}
Simplificar
Gráfica
0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.7502
Trazar el gráfico f(x)
Respuesta [src] 
   /                2\                                                                   2                                       2                          2    /                2\                         
   |    /       ___\ |                    /      ___\          /  ___\       /       ___\  /          /      ___\\   /       ___\     /  ___\   /       ___\     |    /       ___\ |         /       ___\    
log\1 + \-1 + \/ 2 / /          pi*I + log\2 - \/ 2 /       log\\/ 2 /       \-1 + \/ 2 / *\pi*I + log\2 - \/ 2 //   \-1 + \/ 2 / *log\\/ 2 /   \-1 + \/ 2 / *log\1 + \-1 + \/ 2 / /      pi*\-1 + \/ 2 /    
---------------------- - pi*I - --------------------- - ------------------ + ------------------------------------- + ------------------------ - ------------------------------------ - ----------------------
                   2                               2                     2                              2                                2                                2              /                 2\
       /       ___\                    /       ___\          /       ___\                   /       ___\                     /       ___\                     /       ___\               |     /       ___\ |
  -1 + \-1 + \/ 2 /               -1 + \-1 + \/ 2 /     -1 + \-1 + \/ 2 /              -1 + \-1 + \/ 2 /                -1 + \-1 + \/ 2 /                -1 + \-1 + \/ 2 /             2*\-1 + \-1 + \/ 2 / /
log⁡((−1+2)2+1)−1+(−1+2)2+(−1+2)2log⁡(2)−1+(−1+2)2−(−1+2)2log⁡((−1+2)2+1)−1+(−1+2)2−log⁡(2)−1+(−1+2)2−π(−1+2)2(−1+(−1+2)2)−iπ+(−1+2)2(log⁡(2−2)+iπ)−1+(−1+2)2−log⁡(2−2)+iπ−1+(−1+2)2\frac{\log{\left(\left(-1 + \sqrt{2}\right)^{2} + 1 \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} + \frac{\left(-1 + \sqrt{2}\right)^{2} \log{\left(\sqrt{2} \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\left(-1 + \sqrt{2}\right)^{2} \log{\left(\left(-1 + \sqrt{2}\right)^{2} + 1 \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\log{\left(\sqrt{2} \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\pi \left(-1 + \sqrt{2}\right)}{2 \left(-1 + \left(-1 + \sqrt{2}\right)^{2}\right)} - i \pi + \frac{\left(-1 + \sqrt{2}\right)^{2} \left(\log{\left(2 - \sqrt{2} \right)} + i \pi\right)}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\log{\left(2 - \sqrt{2} \right)} + i \pi}{-1 + \left(-1 + \sqrt{2}\right)^{2}} 
=
= 
   /                2\                                                                   2                                       2                          2    /                2\                         
   |    /       ___\ |                    /      ___\          /  ___\       /       ___\  /          /      ___\\   /       ___\     /  ___\   /       ___\     |    /       ___\ |         /       ___\    
log\1 + \-1 + \/ 2 / /          pi*I + log\2 - \/ 2 /       log\\/ 2 /       \-1 + \/ 2 / *\pi*I + log\2 - \/ 2 //   \-1 + \/ 2 / *log\\/ 2 /   \-1 + \/ 2 / *log\1 + \-1 + \/ 2 / /      pi*\-1 + \/ 2 /    
---------------------- - pi*I - --------------------- - ------------------ + ------------------------------------- + ------------------------ - ------------------------------------ - ----------------------
                   2                               2                     2                              2                                2                                2              /                 2\
       /       ___\                    /       ___\          /       ___\                   /       ___\                     /       ___\                     /       ___\               |     /       ___\ |
  -1 + \-1 + \/ 2 /               -1 + \-1 + \/ 2 /     -1 + \-1 + \/ 2 /              -1 + \-1 + \/ 2 /                -1 + \-1 + \/ 2 /                -1 + \-1 + \/ 2 /             2*\-1 + \-1 + \/ 2 / /
log⁡((−1+2)2+1)−1+(−1+2)2+(−1+2)2log⁡(2)−1+(−1+2)2−(−1+2)2log⁡((−1+2)2+1)−1+(−1+2)2−log⁡(2)−1+(−1+2)2−π(−1+2)2(−1+(−1+2)2)−iπ+(−1+2)2(log⁡(2−2)+iπ)−1+(−1+2)2−log⁡(2−2)+iπ−1+(−1+2)2\frac{\log{\left(\left(-1 + \sqrt{2}\right)^{2} + 1 \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} + \frac{\left(-1 + \sqrt{2}\right)^{2} \log{\left(\sqrt{2} \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\left(-1 + \sqrt{2}\right)^{2} \log{\left(\left(-1 + \sqrt{2}\right)^{2} + 1 \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\log{\left(\sqrt{2} \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\pi \left(-1 + \sqrt{2}\right)}{2 \left(-1 + \left(-1 + \sqrt{2}\right)^{2}\right)} - i \pi + \frac{\left(-1 + \sqrt{2}\right)^{2} \left(\log{\left(2 - \sqrt{2} \right)} + i \pi\right)}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\log{\left(2 - \sqrt{2} \right)} + i \pi}{-1 + \left(-1 + \sqrt{2}\right)^{2}} 
log(1 + (-1 + sqrt(2))^2)/(-1 + (-1 + sqrt(2))^2) - pi*i - (pi*i + log(2 - sqrt(2)))/(-1 + (-1 + sqrt(2))^2) - log(sqrt(2))/(-1 + (-1 + sqrt(2))^2) + (-1 + sqrt(2))^2*(pi*i + log(2 - sqrt(2)))/(-1 + (-1 + sqrt(2))^2) + (-1 + sqrt(2))^2*log(sqrt(2))/(-1 + (-1 + sqrt(2))^2) - (-1 + sqrt(2))^2*log(1 + (-1 + sqrt(2))^2)/(-1 + (-1 + sqrt(2))^2) - pi*(-1 + sqrt(2))/(2*(-1 + (-1 + sqrt(2))^2))
Respuesta numérica [src] 
0.438824573117476
0.438824573117476

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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