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Resolución de integrales/ sin(x)/ 1/(3sin(x)+1)

Integral de 1/(3sin(x)+1) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |       1         
 |  ------------ dx
 |  3*sin(x) + 1   
 |                 
/                  
0
$$\int\limits_{0}^{1} \frac{1}{3 \sin{\left(x \right)} + 1}\, dx$$ 
Integral(1/(3*sin(x) + 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                        ___    /        ___      /x\\     ___    /        ___      /x\\
 |                       \/ 2 *log|3 + 2*\/ 2  + tan|-||   \/ 2 *log|3 - 2*\/ 2  + tan|-||
 |      1                         \                 \2//            \                 \2//
 | ------------ dx = C - ------------------------------- + -------------------------------
 | 3*sin(x) + 1                         4                                 4               
 |                                                                                        
/
$$\int \frac{1}{3 \sin{\left(x \right)} + 1}\, dx = C + \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 2 \sqrt{2} + 3 \right)}}{4} - \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} + 2 \sqrt{2} + 3 \right)}}{4}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
    ___    /        ___\     ___    /        ___           \     ___    /        ___\     ___    /        ___           \
  \/ 2 *log\3 - 2*\/ 2 /   \/ 2 *log\3 + 2*\/ 2  + tan(1/2)/   \/ 2 *log\3 + 2*\/ 2 /   \/ 2 *log\3 - 2*\/ 2  + tan(1/2)/
- ---------------------- - --------------------------------- + ---------------------- + ---------------------------------
            4                              4                             4                              4
$$- \frac{\sqrt{2} \log{\left(\tan{\left(\frac{1}{2} \right)} + 2 \sqrt{2} + 3 \right)}}{4} + \frac{\sqrt{2} \log{\left(- 2 \sqrt{2} + \tan{\left(\frac{1}{2} \right)} + 3 \right)}}{4} - \frac{\sqrt{2} \log{\left(3 - 2 \sqrt{2} \right)}}{4} + \frac{\sqrt{2} \log{\left(2 \sqrt{2} + 3 \right)}}{4}$$ 
=
= 
    ___    /        ___\     ___    /        ___           \     ___    /        ___\     ___    /        ___           \
  \/ 2 *log\3 - 2*\/ 2 /   \/ 2 *log\3 + 2*\/ 2  + tan(1/2)/   \/ 2 *log\3 + 2*\/ 2 /   \/ 2 *log\3 - 2*\/ 2  + tan(1/2)/
- ---------------------- - --------------------------------- + ---------------------- + ---------------------------------
            4                              4                             4                              4
$$- \frac{\sqrt{2} \log{\left(\tan{\left(\frac{1}{2} \right)} + 2 \sqrt{2} + 3 \right)}}{4} + \frac{\sqrt{2} \log{\left(- 2 \sqrt{2} + \tan{\left(\frac{1}{2} \right)} + 3 \right)}}{4} - \frac{\sqrt{2} \log{\left(3 - 2 \sqrt{2} \right)}}{4} + \frac{\sqrt{2} \log{\left(2 \sqrt{2} + 3 \right)}}{4}$$ 
-sqrt(2)*log(3 - 2*sqrt(2))/4 - sqrt(2)*log(3 + 2*sqrt(2) + tan(1/2))/4 + sqrt(2)*log(3 + 2*sqrt(2))/4 + sqrt(2)*log(3 - 2*sqrt(2) + tan(1/2))/4
Respuesta numérica [src] 
0.474360236604794
0.474360236604794

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

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