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Resolución de integrales/ sin^2/ dx/(3+sin^2(x))

Integral de dx/(3+sin^2(x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi               
 --               
 4                
  /               
 |                
 |       1        
 |  ----------- dx
 |         2      
 |  3 + sin (x)   
 |                
/                 
0
$$\int\limits_{0}^{\frac{\pi}{4}} \frac{1}{\sin^{2}{\left(x \right)} + 3}\, dx$$ 
Integral(1/(3 + sin(x)^2), (x, 0, pi/4))
Respuesta (Indefinida) [src] 
                              /        /x   pi\                     \         /        /x   pi\       /  ___    /x\\\
                              |        |- - --|                     |         |        |- - --|       |\/ 3 *tan|-|||
  /                       ___ |        |2   2 |       /  ___    /x\\|     ___ |        |2   2 |       |         \2/||
 |                      \/ 3 *|pi*floor|------| + atan|\/ 3 *tan|-|||   \/ 3 *|pi*floor|------| + atan|------------||
 |      1                     \        \  pi  /       \         \2///         \        \  pi  /       \     3      //
 | ----------- dx = C + --------------------------------------------- + ---------------------------------------------
 |        2                                   6                                               6                      
 | 3 + sin (x)                                                                                                       
 |                                                                                                                   
/
$$\int \frac{1}{\sin^{2}{\left(x \right)} + 3}\, dx = C + \frac{\sqrt{3} \left(\operatorname{atan}{\left(\frac{\sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{6} + \frac{\sqrt{3} \left(\operatorname{atan}{\left(\sqrt{3} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{6}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                                          /          /  ___ /       ___\\\
                                                      ___ |          |\/ 3 *\-1 + \/ 2 /||
     ___     ___ /          /  ___ /       ___\\\   \/ 3 *|-pi + atan|------------------||
pi*\/ 3    \/ 3 *\-pi + atan\\/ 3 *\-1 + \/ 2 ///         \          \        3         //
-------- + -------------------------------------- + --------------------------------------
   3                         6                                        6
$$\frac{\sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3} \left(-1 + \sqrt{2}\right)}{3} \right)}\right)}{6} + \frac{\sqrt{3} \left(- \pi + \operatorname{atan}{\left(\sqrt{3} \left(-1 + \sqrt{2}\right) \right)}\right)}{6} + \frac{\sqrt{3} \pi}{3}$$ 
=
= 
                                                          /          /  ___ /       ___\\\
                                                      ___ |          |\/ 3 *\-1 + \/ 2 /||
     ___     ___ /          /  ___ /       ___\\\   \/ 3 *|-pi + atan|------------------||
pi*\/ 3    \/ 3 *\-pi + atan\\/ 3 *\-1 + \/ 2 ///         \          \        3         //
-------- + -------------------------------------- + --------------------------------------
   3                         6                                        6
$$\frac{\sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3} \left(-1 + \sqrt{2}\right)}{3} \right)}\right)}{6} + \frac{\sqrt{3} \left(- \pi + \operatorname{atan}{\left(\sqrt{3} \left(-1 + \sqrt{2}\right) \right)}\right)}{6} + \frac{\sqrt{3} \pi}{3}$$ 
pi*sqrt(3)/3 + sqrt(3)*(-pi + atan(sqrt(3)*(-1 + sqrt(2))))/6 + sqrt(3)*(-pi + atan(sqrt(3)*(-1 + sqrt(2))/3))/6
Respuesta numérica [src] 
0.247415359903075
0.247415359903075

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

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