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Resolución de integrales/ (-4(cos(x))^2*sin(x))/(cos(x)+sin(x))

Integral de (-4(cos(x))^2*sin(x))/(cos(x)+sin(x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                     
  /                     
 |                      
 |        2             
 |  -4*cos (x)*sin(x)   
 |  ----------------- dx
 |   cos(x) + sin(x)    
 |                      
/                       
0
∫01sin⁡(x)(−4cos⁡2(x))sin⁡(x)+cos⁡(x) dx\int\limits_{0}^{1} \frac{\sin{\left(x \right)} \left(- 4 \cos^{2}{\left(x \right)}\right)}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx 
Integral(((-4*cos(x)^2)*sin(x))/(cos(x) + sin(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                                                                                                                                                                                                                                                  
 |                                          2/x\                        /x\                    /       2/x\\           /       2/x\        /x\\                3/x\                2/x\    /       2/x\\        4/x\    /       2/x\\        4/x\    /       2/x\        /x\\        2/x\    /       2/x\        /x\\
 |       2                            16*tan |-|                   8*tan|-|               4*log|1 + tan |-||      4*log|1 - tan |-| + 2*tan|-||           8*tan |-|           8*tan |-|*log|1 + tan |-||   4*tan |-|*log|1 + tan |-||   4*tan |-|*log|1 - tan |-| + 2*tan|-||   8*tan |-|*log|1 - tan |-| + 2*tan|-||
 | -4*cos (x)*sin(x)                         \2/                        \2/                    \        \2//           \        \2/        \2//                 \2/                 \2/    \        \2//         \2/    \        \2//         \2/    \        \2/        \2//         \2/    \        \2/        \2//
 | ----------------- dx = C - ------------------------- - ------------------------- - ------------------------- + ----------------------------- + ------------------------- - -------------------------- - -------------------------- + ------------------------------------- + -------------------------------------
 |  cos(x) + sin(x)                    4/x\        2/x\            4/x\        2/x\            4/x\        2/x\              4/x\        2/x\              4/x\        2/x\            4/x\        2/x\             4/x\        2/x\                   4/x\        2/x\                        4/x\        2/x\      
 |                            4 + 4*tan |-| + 8*tan |-|   4 + 4*tan |-| + 8*tan |-|   4 + 4*tan |-| + 8*tan |-|     4 + 4*tan |-| + 8*tan |-|     4 + 4*tan |-| + 8*tan |-|   4 + 4*tan |-| + 8*tan |-|    4 + 4*tan |-| + 8*tan |-|          4 + 4*tan |-| + 8*tan |-|               4 + 4*tan |-| + 8*tan |-|      
/                                       \2/         \2/             \2/         \2/             \2/         \2/               \2/         \2/               \2/         \2/             \2/         \2/              \2/         \2/                    \2/         \2/                         \2/         \2/
∫sin⁡(x)(−4cos⁡2(x))sin⁡(x)+cos⁡(x) dx=C−4log⁡(tan⁡2(x2)+1)tan⁡4(x2)4tan⁡4(x2)+8tan⁡2(x2)+4−8log⁡(tan⁡2(x2)+1)tan⁡2(x2)4tan⁡4(x2)+8tan⁡2(x2)+4−4log⁡(tan⁡2(x2)+1)4tan⁡4(x2)+8tan⁡2(x2)+4+4log⁡(−tan⁡2(x2)+2tan⁡(x2)+1)tan⁡4(x2)4tan⁡4(x2)+8tan⁡2(x2)+4+8log⁡(−tan⁡2(x2)+2tan⁡(x2)+1)tan⁡2(x2)4tan⁡4(x2)+8tan⁡2(x2)+4+4log⁡(−tan⁡2(x2)+2tan⁡(x2)+1)4tan⁡4(x2)+8tan⁡2(x2)+4+8tan⁡3(x2)4tan⁡4(x2)+8tan⁡2(x2)+4−16tan⁡2(x2)4tan⁡4(x2)+8tan⁡2(x2)+4−8tan⁡(x2)4tan⁡4(x2)+8tan⁡2(x2)+4\int \frac{\sin{\left(x \right)} \left(- 4 \cos^{2}{\left(x \right)}\right)}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx = C - \frac{4 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{8 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{4 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{4 \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{8 \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{4 \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{8 \tan^{3}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{16 \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{8 \tan{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.901-2
Trazar el gráfico f(x)
Respuesta [src] 
                 2                                                          /       2     \            /       2                  \                 3                      2         /       2     \        4         /       2     \        4         /       2                  \        2         /       2                  \
           16*tan (1/2)                     8*tan(1/2)                 4*log\1 + tan (1/2)/       4*log\1 - tan (1/2) + 2*tan(1/2)/            8*tan (1/2)            8*tan (1/2)*log\1 + tan (1/2)/   4*tan (1/2)*log\1 + tan (1/2)/   4*tan (1/2)*log\1 - tan (1/2) + 2*tan(1/2)/   8*tan (1/2)*log\1 - tan (1/2) + 2*tan(1/2)/
- ----------------------------- - ----------------------------- - ----------------------------- + --------------------------------- + ----------------------------- - ------------------------------ - ------------------------------ + ------------------------------------------- + -------------------------------------------
           4             2                 4             2                 4             2                   4             2                   4             2                 4             2                  4             2                         4             2                               4             2            
  4 + 4*tan (1/2) + 8*tan (1/2)   4 + 4*tan (1/2) + 8*tan (1/2)   4 + 4*tan (1/2) + 8*tan (1/2)     4 + 4*tan (1/2) + 8*tan (1/2)     4 + 4*tan (1/2) + 8*tan (1/2)   4 + 4*tan (1/2) + 8*tan (1/2)    4 + 4*tan (1/2) + 8*tan (1/2)           4 + 4*tan (1/2) + 8*tan (1/2)                 4 + 4*tan (1/2) + 8*tan (1/2)
−16tan⁡2(12)4tan⁡4(12)+8tan⁡2(12)+4−8tan⁡(12)4tan⁡4(12)+8tan⁡2(12)+4−4log⁡(tan⁡2(12)+1)4tan⁡4(12)+8tan⁡2(12)+4−8log⁡(tan⁡2(12)+1)tan⁡2(12)4tan⁡4(12)+8tan⁡2(12)+4−4log⁡(tan⁡2(12)+1)tan⁡4(12)4tan⁡4(12)+8tan⁡2(12)+4+4log⁡(−tan⁡2(12)+1+2tan⁡(12))tan⁡4(12)4tan⁡4(12)+8tan⁡2(12)+4+8tan⁡3(12)4tan⁡4(12)+8tan⁡2(12)+4+8log⁡(−tan⁡2(12)+1+2tan⁡(12))tan⁡2(12)4tan⁡4(12)+8tan⁡2(12)+4+4log⁡(−tan⁡2(12)+1+2tan⁡(12))4tan⁡4(12)+8tan⁡2(12)+4- \frac{16 \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{8 \tan{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{4 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{8 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{4 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{4 \log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{8 \tan^{3}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{8 \log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{4 \log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} 
=
= 
                 2                                                          /       2     \            /       2                  \                 3                      2         /       2     \        4         /       2     \        4         /       2                  \        2         /       2                  \
           16*tan (1/2)                     8*tan(1/2)                 4*log\1 + tan (1/2)/       4*log\1 - tan (1/2) + 2*tan(1/2)/            8*tan (1/2)            8*tan (1/2)*log\1 + tan (1/2)/   4*tan (1/2)*log\1 + tan (1/2)/   4*tan (1/2)*log\1 - tan (1/2) + 2*tan(1/2)/   8*tan (1/2)*log\1 - tan (1/2) + 2*tan(1/2)/
- ----------------------------- - ----------------------------- - ----------------------------- + --------------------------------- + ----------------------------- - ------------------------------ - ------------------------------ + ------------------------------------------- + -------------------------------------------
           4             2                 4             2                 4             2                   4             2                   4             2                 4             2                  4             2                         4             2                               4             2            
  4 + 4*tan (1/2) + 8*tan (1/2)   4 + 4*tan (1/2) + 8*tan (1/2)   4 + 4*tan (1/2) + 8*tan (1/2)     4 + 4*tan (1/2) + 8*tan (1/2)     4 + 4*tan (1/2) + 8*tan (1/2)   4 + 4*tan (1/2) + 8*tan (1/2)    4 + 4*tan (1/2) + 8*tan (1/2)           4 + 4*tan (1/2) + 8*tan (1/2)                 4 + 4*tan (1/2) + 8*tan (1/2)
−16tan⁡2(12)4tan⁡4(12)+8tan⁡2(12)+4−8tan⁡(12)4tan⁡4(12)+8tan⁡2(12)+4−4log⁡(tan⁡2(12)+1)4tan⁡4(12)+8tan⁡2(12)+4−8log⁡(tan⁡2(12)+1)tan⁡2(12)4tan⁡4(12)+8tan⁡2(12)+4−4log⁡(tan⁡2(12)+1)tan⁡4(12)4tan⁡4(12)+8tan⁡2(12)+4+4log⁡(−tan⁡2(12)+1+2tan⁡(12))tan⁡4(12)4tan⁡4(12)+8tan⁡2(12)+4+8tan⁡3(12)4tan⁡4(12)+8tan⁡2(12)+4+8log⁡(−tan⁡2(12)+1+2tan⁡(12))tan⁡2(12)4tan⁡4(12)+8tan⁡2(12)+4+4log⁡(−tan⁡2(12)+1+2tan⁡(12))4tan⁡4(12)+8tan⁡2(12)+4- \frac{16 \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{8 \tan{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{4 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{8 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{4 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{4 \log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{8 \tan^{3}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{8 \log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{4 \log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} 
-16*tan(1/2)^2/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) - 8*tan(1/2)/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) - 4*log(1 + tan(1/2)^2)/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) + 4*log(1 - tan(1/2)^2 + 2*tan(1/2))/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) + 8*tan(1/2)^3/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) - 8*tan(1/2)^2*log(1 + tan(1/2)^2)/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) - 4*tan(1/2)^4*log(1 + tan(1/2)^2)/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) + 4*tan(1/2)^4*log(1 - tan(1/2)^2 + 2*tan(1/2))/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) + 8*tan(1/2)^2*log(1 - tan(1/2)^2 + 2*tan(1/2))/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2)
Respuesta numérica [src] 
-0.83935446417103
-0.83935446417103

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

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