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Integral de ax/cos^2x dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  0           
  /           
 |            
 |    a*x     
 |  ------- dx
 |     2      
 |  cos (x)   
 |            
/             
pi            
--            
4             
$$\int\limits_{\frac{\pi}{4}}^{0} \frac{a x}{\cos^{2}{\left(x \right)}}\, dx$$
Integral((a*x)/cos(x)^2, (x, pi/4, 0))
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|-| |
 |   a*x              |   \        \2//      \       \2//      \        \2//       \2/    \       \2//       \2/    \        \2//       \2/    \        \2//           \2/ |
 | ------- dx = C + a*|---------------- - --------------- - ---------------- + ----------------------- + ------------------------ - ------------------------ - ------------|
 |    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//
/                                                                                                                                                                           
$$\int \frac{a x}{\cos^{2}{\left(x \right)}}\, dx = C + a \left(- \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}\right)$$
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 /    |         
- a*|---------------------- - --------------------- - ------------------ + ------------------------------------- + ------------------------ - ------------------------------------ - ----------------------| + pi*I*a
    |                   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 / //         
$$- a \left(\frac{\log{\left(\left(-1 + \sqrt{2}\right)^{2} + 1 \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} + \frac{\left(-1 + \sqrt{2}\right)^{2} \log{\left(\sqrt{2} \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\left(-1 + \sqrt{2}\right)^{2} \log{\left(\left(-1 + \sqrt{2}\right)^{2} + 1 \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\log{\left(\sqrt{2} \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\pi \left(-1 + \sqrt{2}\right)}{2 \left(-1 + \left(-1 + \sqrt{2}\right)^{2}\right)} + \frac{\left(-1 + \sqrt{2}\right)^{2} \left(\log{\left(2 - \sqrt{2} \right)} + i \pi\right)}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\log{\left(2 - \sqrt{2} \right)} + i \pi}{-1 + \left(-1 + \sqrt{2}\right)^{2}}\right) + i \pi a$$
=
=
    /   /                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 /    |         
- a*|---------------------- - --------------------- - ------------------ + ------------------------------------- + ------------------------ - ------------------------------------ - ----------------------| + pi*I*a
    |                   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 / //         
$$- a \left(\frac{\log{\left(\left(-1 + \sqrt{2}\right)^{2} + 1 \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} + \frac{\left(-1 + \sqrt{2}\right)^{2} \log{\left(\sqrt{2} \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\left(-1 + \sqrt{2}\right)^{2} \log{\left(\left(-1 + \sqrt{2}\right)^{2} + 1 \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\log{\left(\sqrt{2} \right)}}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\pi \left(-1 + \sqrt{2}\right)}{2 \left(-1 + \left(-1 + \sqrt{2}\right)^{2}\right)} + \frac{\left(-1 + \sqrt{2}\right)^{2} \left(\log{\left(2 - \sqrt{2} \right)} + i \pi\right)}{-1 + \left(-1 + \sqrt{2}\right)^{2}} - \frac{\log{\left(2 - \sqrt{2} \right)} + i \pi}{-1 + \left(-1 + \sqrt{2}\right)^{2}}\right) + i \pi a$$
-a*(log(1 + (-1 + sqrt(2))^2)/(-1 + (-1 + sqrt(2))^2) - (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))) + pi*i*a

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