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Resolución de integrales/ sin^2/ 1/(1+sin^2x)

Integral de 1/(1+sin^2x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |       1        
 |  ----------- dx
 |         2      
 |  1 + sin (x)   
 |                
/                 
0
$$\int\limits_{0}^{1} \frac{1}{\sin^{2}{\left(x \right)} + 1}\, dx$$ 
Integral(1/(1 + sin(x)^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                          /        /x   pi\       /        /x\     \\            /        /x   pi\       /        /x\     \\                                       /        /x   pi\       /        /x\     \\                                             /        /x   pi\       /        /x\     \\
                          |        |- - --|       |     tan|-|     ||            |        |- - --|       |     tan|-|     ||        _____________    _____________ |        |- - --|       |     tan|-|     ||              _____________    _____________ |        |- - --|       |     tan|-|     ||
                          |        |2   2 |       |        \2/     ||        ___ |        |2   2 |       |        \2/     ||       /         ___    /         ___  |        |2   2 |       |        \2/     ||       ___   /         ___    /         ___  |        |2   2 |       |        \2/     ||
                        3*|pi*floor|------| + atan|----------------||    2*\/ 2 *|pi*floor|------| + atan|----------------||   3*\/  3 - 2*\/ 2  *\/  3 + 2*\/ 2  *|pi*floor|------| + atan|----------------||   2*\/ 2 *\/  3 - 2*\/ 2  *\/  3 + 2*\/ 2  *|pi*floor|------| + atan|----------------||
  /                       |        \  pi  /       |   _____________||            |        \  pi  /       |   _____________||                                       |        \  pi  /       |   _____________||                                             |        \  pi  /       |   _____________||
 |                        |                       |  /         ___ ||            |                       |  /         ___ ||                                       |                       |  /         ___ ||                                             |                       |  /         ___ ||
 |      1                 \                       \\/  3 - 2*\/ 2  //            \                       \\/  3 - 2*\/ 2  //                                       \                       \\/  3 + 2*\/ 2  //                                             \                       \\/  3 + 2*\/ 2  //
 | ----------- dx = C + ---------------------------------------------- + --------------------------------------------------- + ------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------
 |        2                   _____________              _____________            _____________              _____________                            _____________              _____________                                             _____________              _____________                   
 | 1 + sin (x)               /         ___        ___   /         ___            /         ___        ___   /         ___                            /         ___        ___   /         ___                                             /         ___        ___   /         ___                    
 |                      10*\/  3 - 2*\/ 2   + 7*\/ 2 *\/  3 - 2*\/ 2        10*\/  3 - 2*\/ 2   + 7*\/ 2 *\/  3 - 2*\/ 2                        10*\/  3 - 2*\/ 2   + 7*\/ 2 *\/  3 - 2*\/ 2                                         10*\/  3 - 2*\/ 2   + 7*\/ 2 *\/  3 - 2*\/ 2                     
/
$$\int \frac{1}{\sin^{2}{\left(x \right)} + 1}\, dx = C + \frac{2 \sqrt{2} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{2 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                                                                                                                             _____________    _____________                               
                                                                /    tan(1/2)    \                                                          /         ___    /         ___  /          /    tan(1/2)    \\
                                                      -pi + atan|----------------|                                                        \/  3 - 2*\/ 2  *\/  3 + 2*\/ 2  *|-pi + atan|----------------||
                                                                |   _____________|                    _____________    _____________                                        |          |   _____________||
                                                                |  /         ___ |                   /         ___    /         ___                                         |          |  /         ___ ||
                     pi                                         \\/  3 - 2*\/ 2  /              pi*\/  3 - 2*\/ 2  *\/  3 + 2*\/ 2                                          \          \\/  3 + 2*\/ 2  //
------------------------------------------- + ------------------------------------------- + ------------------------------------------- + ----------------------------------------------------------------
     _____________            _____________        _____________            _____________        _____________            _____________                  _____________            _____________           
    /         ___      ___   /         ___        /         ___      ___   /         ___        /         ___      ___   /         ___                  /         ___      ___   /         ___            
2*\/  3 - 2*\/ 2   + \/ 2 *\/  3 - 2*\/ 2     2*\/  3 - 2*\/ 2   + \/ 2 *\/  3 - 2*\/ 2     2*\/  3 - 2*\/ 2   + \/ 2 *\/  3 - 2*\/ 2               2*\/  3 - 2*\/ 2   + \/ 2 *\/  3 - 2*\/ 2
$$\frac{\sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \left(- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}\right)}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}} + \frac{- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}} + \frac{\pi}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}} + \frac{\pi \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}}$$ 
=
= 
                                                                                                                                             _____________    _____________                               
                                                                /    tan(1/2)    \                                                          /         ___    /         ___  /          /    tan(1/2)    \\
                                                      -pi + atan|----------------|                                                        \/  3 - 2*\/ 2  *\/  3 + 2*\/ 2  *|-pi + atan|----------------||
                                                                |   _____________|                    _____________    _____________                                        |          |   _____________||
                                                                |  /         ___ |                   /         ___    /         ___                                         |          |  /         ___ ||
                     pi                                         \\/  3 - 2*\/ 2  /              pi*\/  3 - 2*\/ 2  *\/  3 + 2*\/ 2                                          \          \\/  3 + 2*\/ 2  //
------------------------------------------- + ------------------------------------------- + ------------------------------------------- + ----------------------------------------------------------------
     _____________            _____________        _____________            _____________        _____________            _____________                  _____________            _____________           
    /         ___      ___   /         ___        /         ___      ___   /         ___        /         ___      ___   /         ___                  /         ___      ___   /         ___            
2*\/  3 - 2*\/ 2   + \/ 2 *\/  3 - 2*\/ 2     2*\/  3 - 2*\/ 2   + \/ 2 *\/  3 - 2*\/ 2     2*\/  3 - 2*\/ 2   + \/ 2 *\/  3 - 2*\/ 2               2*\/  3 - 2*\/ 2   + \/ 2 *\/  3 - 2*\/ 2
$$\frac{\sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \left(- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}\right)}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}} + \frac{- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}} + \frac{\pi}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}} + \frac{\pi \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}}$$ 
pi/(2*sqrt(3 - 2*sqrt(2)) + sqrt(2)*sqrt(3 - 2*sqrt(2))) + (-pi + atan(tan(1/2)/sqrt(3 - 2*sqrt(2))))/(2*sqrt(3 - 2*sqrt(2)) + sqrt(2)*sqrt(3 - 2*sqrt(2))) + pi*sqrt(3 - 2*sqrt(2))*sqrt(3 + 2*sqrt(2))/(2*sqrt(3 - 2*sqrt(2)) + sqrt(2)*sqrt(3 - 2*sqrt(2))) + sqrt(3 - 2*sqrt(2))*sqrt(3 + 2*sqrt(2))*(-pi + atan(tan(1/2)/sqrt(3 + 2*sqrt(2))))/(2*sqrt(3 - 2*sqrt(2)) + sqrt(2)*sqrt(3 - 2*sqrt(2)))
Respuesta numérica [src] 
0.809352817335295
0.809352817335295

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

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