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Resolución de integrales/ cos²x/ sinx/(sqrcos²x+1)

Integral de sinx/(sqrcos²x+1) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |     sin(x)     
 |  ----------- dx
 |     4          
 |  cos (x) + 1   
 |                
/                 
0
$$\int\limits_{0}^{1} \frac{\sin{\left(x \right)}}{\cos^{4}{\left(x \right)} + 1}\, dx$$ 
Integral(sin(x)/(cos(x)^4 + 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                                                                                                         
 |                        ___     /      ___       \     ___     /       ___       \     ___    /         2          ___       \     ___    /         2          ___       \
 |    sin(x)            \/ 2 *atan\1 + \/ 2 *cos(x)/   \/ 2 *atan\-1 + \/ 2 *cos(x)/   \/ 2 *log\4 + 4*cos (x) + 4*\/ 2 *cos(x)/   \/ 2 *log\4 + 4*cos (x) - 4*\/ 2 *cos(x)/
 | ----------- dx = C - ---------------------------- - ----------------------------- - ----------------------------------------- + -----------------------------------------
 |    4                              4                               4                                     8                                           8                    
 | cos (x) + 1                                                                                                                                                              
 |                                                                                                                                                                          
/
$$\int \frac{\sin{\left(x \right)}}{\cos^{4}{\left(x \right)} + 1}\, dx = C + \frac{\sqrt{2} \log{\left(4 \cos^{2}{\left(x \right)} - 4 \sqrt{2} \cos{\left(x \right)} + 4 \right)}}{8} - \frac{\sqrt{2} \log{\left(4 \cos^{2}{\left(x \right)} + 4 \sqrt{2} \cos{\left(x \right)} + 4 \right)}}{8} - \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} \cos{\left(x \right)} - 1 \right)}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} \cos{\left(x \right)} + 1 \right)}}{4}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
    ___     /      ___       \     ___    /        ___\     ___    /         2          ___       \     ___     /      ___       \        ___     ___    /        ___\     ___    /         2          ___       \
  \/ 2 *atan\1 + \/ 2 *cos(1)/   \/ 2 *log\8 - 4*\/ 2 /   \/ 2 *log\4 + 4*cos (1) + 4*\/ 2 *cos(1)/   \/ 2 *atan\1 - \/ 2 *cos(1)/   pi*\/ 2    \/ 2 *log\8 + 4*\/ 2 /   \/ 2 *log\4 + 4*cos (1) - 4*\/ 2 *cos(1)/
- ---------------------------- - ---------------------- - ----------------------------------------- + ---------------------------- + -------- + ---------------------- + -----------------------------------------
               4                           8                                  8                                    4                    8                 8                                  8
$$- \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} \cos{\left(1 \right)} + 1 \right)}}{4} - \frac{\sqrt{2} \log{\left(4 \cos^{2}{\left(1 \right)} + 4 \sqrt{2} \cos{\left(1 \right)} + 4 \right)}}{8} - \frac{\sqrt{2} \log{\left(8 - 4 \sqrt{2} \right)}}{8} + \frac{\sqrt{2} \operatorname{atan}{\left(- \sqrt{2} \cos{\left(1 \right)} + 1 \right)}}{4} + \frac{\sqrt{2} \log{\left(- 4 \sqrt{2} \cos{\left(1 \right)} + 4 \cos^{2}{\left(1 \right)} + 4 \right)}}{8} + \frac{\sqrt{2} \log{\left(4 \sqrt{2} + 8 \right)}}{8} + \frac{\sqrt{2} \pi}{8}$$ 
=
= 
    ___     /      ___       \     ___    /        ___\     ___    /         2          ___       \     ___     /      ___       \        ___     ___    /        ___\     ___    /         2          ___       \
  \/ 2 *atan\1 + \/ 2 *cos(1)/   \/ 2 *log\8 - 4*\/ 2 /   \/ 2 *log\4 + 4*cos (1) + 4*\/ 2 *cos(1)/   \/ 2 *atan\1 - \/ 2 *cos(1)/   pi*\/ 2    \/ 2 *log\8 + 4*\/ 2 /   \/ 2 *log\4 + 4*cos (1) - 4*\/ 2 *cos(1)/
- ---------------------------- - ---------------------- - ----------------------------------------- + ---------------------------- + -------- + ---------------------- + -----------------------------------------
               4                           8                                  8                                    4                    8                 8                                  8
$$- \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} \cos{\left(1 \right)} + 1 \right)}}{4} - \frac{\sqrt{2} \log{\left(4 \cos^{2}{\left(1 \right)} + 4 \sqrt{2} \cos{\left(1 \right)} + 4 \right)}}{8} - \frac{\sqrt{2} \log{\left(8 - 4 \sqrt{2} \right)}}{8} + \frac{\sqrt{2} \operatorname{atan}{\left(- \sqrt{2} \cos{\left(1 \right)} + 1 \right)}}{4} + \frac{\sqrt{2} \log{\left(- 4 \sqrt{2} \cos{\left(1 \right)} + 4 \cos^{2}{\left(1 \right)} + 4 \right)}}{8} + \frac{\sqrt{2} \log{\left(4 \sqrt{2} + 8 \right)}}{8} + \frac{\sqrt{2} \pi}{8}$$ 
-sqrt(2)*atan(1 + sqrt(2)*cos(1))/4 - sqrt(2)*log(8 - 4*sqrt(2))/8 - sqrt(2)*log(4 + 4*cos(1)^2 + 4*sqrt(2)*cos(1))/8 + sqrt(2)*atan(1 - sqrt(2)*cos(1))/4 + pi*sqrt(2)/8 + sqrt(2)*log(8 + 4*sqrt(2))/8 + sqrt(2)*log(4 + 4*cos(1)^2 - 4*sqrt(2)*cos(1))/8
Respuesta numérica [src] 
0.335467868939454
0.335467868939454

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

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