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Resolución de integrales/ cos^3/ xsinx/ xsinx/cos^3x

Integral de xsinx/cos^3x dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi            
 --            
 4             
  /            
 |             
 |  x*sin(x)   
 |  -------- dx
 |     3       
 |  cos (x)    
 |             
/              
0
∫0π4xsin⁡(x)cos⁡3(x) dx\int\limits_{0}^{\frac{\pi}{4}} \frac{x \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}\, dx 
Integral((x*sin(x))/cos(x)^3, (x, 0, pi/4))
Respuesta (Indefinida) [src] 
  /                                                            /x\                        3/x\                        4/x\                         2/x\       
 |                                                        2*tan|-|                   2*tan |-|                   x*tan |-|                  2*x*tan |-|       
 | x*sin(x)                      x                             \2/                         \2/                         \2/                          \2/       
 | -------- dx = C + ------------------------- - ------------------------- + ------------------------- + ------------------------- + -------------------------
 |    3                       2/x\        4/x\            2/x\        4/x\            2/x\        4/x\            2/x\        4/x\            2/x\        4/x\
 | cos (x)           2 - 4*tan |-| + 2*tan |-|   2 - 4*tan |-| + 2*tan |-|   2 - 4*tan |-| + 2*tan |-|   2 - 4*tan |-| + 2*tan |-|   2 - 4*tan |-| + 2*tan |-|
 |                             \2/         \2/             \2/         \2/             \2/         \2/             \2/         \2/             \2/         \2/
/
∫xsin⁡(x)cos⁡3(x) dx=C+xtan⁡4(x2)2tan⁡4(x2)−4tan⁡2(x2)+2+2xtan⁡2(x2)2tan⁡4(x2)−4tan⁡2(x2)+2+x2tan⁡4(x2)−4tan⁡2(x2)+2+2tan⁡3(x2)2tan⁡4(x2)−4tan⁡2(x2)+2−2tan⁡(x2)2tan⁡4(x2)−4tan⁡2(x2)+2\int \frac{x \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}}\, dx = C + \frac{x \tan^{4}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{2 x \tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{x}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{2 \tan^{3}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2}
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] 
                                                                   3                                                                                      2                                           4            
                /       ___\                           /       ___\                                                                           /       ___\                                /       ___\             
              2*\-1 + \/ 2 /                         2*\-1 + \/ 2 /                                   pi                                   pi*\-1 + \/ 2 /                             pi*\-1 + \/ 2 /             
- ------------------------------------- + ------------------------------------- + ----------------------------------------- + ----------------------------------------- + -----------------------------------------
                    2                 4                     2                 4     /                  2                 4\     /                  2                 4\     /                  2                 4\
        /       ___\      /       ___\          /       ___\      /       ___\      |      /       ___\      /       ___\ |     |      /       ___\      /       ___\ |     |      /       ___\      /       ___\ |
  2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 /    2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 /    4*\2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 / /   2*\2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 / /   4*\2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 / /
−2(−1+2)−4(−1+2)2+2(−1+2)4+2+π(−1+2)44(−4(−1+2)2+2(−1+2)4+2)+2(−1+2)3−4(−1+2)2+2(−1+2)4+2+π(−1+2)22(−4(−1+2)2+2(−1+2)4+2)+π4(−4(−1+2)2+2(−1+2)4+2)- \frac{2 \left(-1 + \sqrt{2}\right)}{- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2} + \frac{\pi \left(-1 + \sqrt{2}\right)^{4}}{4 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)} + \frac{2 \left(-1 + \sqrt{2}\right)^{3}}{- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2} + \frac{\pi \left(-1 + \sqrt{2}\right)^{2}}{2 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)} + \frac{\pi}{4 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)} 
=
= 
                                                                   3                                                                                      2                                           4            
                /       ___\                           /       ___\                                                                           /       ___\                                /       ___\             
              2*\-1 + \/ 2 /                         2*\-1 + \/ 2 /                                   pi                                   pi*\-1 + \/ 2 /                             pi*\-1 + \/ 2 /             
- ------------------------------------- + ------------------------------------- + ----------------------------------------- + ----------------------------------------- + -----------------------------------------
                    2                 4                     2                 4     /                  2                 4\     /                  2                 4\     /                  2                 4\
        /       ___\      /       ___\          /       ___\      /       ___\      |      /       ___\      /       ___\ |     |      /       ___\      /       ___\ |     |      /       ___\      /       ___\ |
  2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 /    2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 /    4*\2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 / /   2*\2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 / /   4*\2 - 4*\-1 + \/ 2 /  + 2*\-1 + \/ 2 / /
−2(−1+2)−4(−1+2)2+2(−1+2)4+2+π(−1+2)44(−4(−1+2)2+2(−1+2)4+2)+2(−1+2)3−4(−1+2)2+2(−1+2)4+2+π(−1+2)22(−4(−1+2)2+2(−1+2)4+2)+π4(−4(−1+2)2+2(−1+2)4+2)- \frac{2 \left(-1 + \sqrt{2}\right)}{- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2} + \frac{\pi \left(-1 + \sqrt{2}\right)^{4}}{4 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)} + \frac{2 \left(-1 + \sqrt{2}\right)^{3}}{- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2} + \frac{\pi \left(-1 + \sqrt{2}\right)^{2}}{2 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)} + \frac{\pi}{4 \left(- 4 \left(-1 + \sqrt{2}\right)^{2} + 2 \left(-1 + \sqrt{2}\right)^{4} + 2\right)} 
-2*(-1 + sqrt(2))/(2 - 4*(-1 + sqrt(2))^2 + 2*(-1 + sqrt(2))^4) + 2*(-1 + sqrt(2))^3/(2 - 4*(-1 + sqrt(2))^2 + 2*(-1 + sqrt(2))^4) + pi/(4*(2 - 4*(-1 + sqrt(2))^2 + 2*(-1 + sqrt(2))^4)) + pi*(-1 + sqrt(2))^2/(2*(2 - 4*(-1 + sqrt(2))^2 + 2*(-1 + sqrt(2))^4)) + pi*(-1 + sqrt(2))^4/(4*(2 - 4*(-1 + sqrt(2))^2 + 2*(-1 + sqrt(2))^4))
Respuesta numérica [src] 
0.285398163397448
0.285398163397448

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

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