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Resolución de integrales/ sin^2/ 1/(1+3sin^2x)

Integral de 1/(1+3sin^2x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                 
  /                 
 |                  
 |        1         
 |  ------------- dx
 |           2      
 |  1 + 3*sin (x)   
 |                  
/                   
0
∫0113sin⁡2(x)+1 dx\int\limits_{0}^{1} \frac{1}{3 \sin^{2}{\left(x \right)} + 1}\, dx 
Integral(1/(1 + 3*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/     ||
                           7*|pi*floor|------| + atan|----------------||    4*\/ 3 *|pi*floor|------| + atan|----------------||   7*\/  7 - 4*\/ 3  *\/  7 + 4*\/ 3  *|pi*floor|------| + atan|----------------||   4*\/ 3 *\/  7 - 4*\/ 3  *\/  7 + 4*\/ 3  *|pi*floor|------| + atan|----------------||
  /                          |        \  pi  /       |   _____________||            |        \  pi  /       |   _____________||                                       |        \  pi  /       |   _____________||                                             |        \  pi  /       |   _____________||
 |                           |                       |  /         ___ ||            |                       |  /         ___ ||                                       |                       |  /         ___ ||                                             |                       |  /         ___ ||
 |       1                   \                       \\/  7 - 4*\/ 3  //            \                       \\/  7 - 4*\/ 3  //                                       \                       \\/  7 + 4*\/ 3  //                                             \                       \\/  7 + 4*\/ 3  //
 | ------------- dx = C + ----------------------------------------------- + --------------------------------------------------- + ------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------
 |          2                   _____________               _____________           _____________               _____________                           _____________               _____________                                            _____________               _____________                   
 | 1 + 3*sin (x)               /         ___         ___   /         ___           /         ___         ___   /         ___                           /         ___         ___   /         ___                                            /         ___         ___   /         ___                    
 |                        52*\/  7 - 4*\/ 3   + 30*\/ 3 *\/  7 - 4*\/ 3       52*\/  7 - 4*\/ 3   + 30*\/ 3 *\/  7 - 4*\/ 3                       52*\/  7 - 4*\/ 3   + 30*\/ 3 *\/  7 - 4*\/ 3                                        52*\/  7 - 4*\/ 3   + 30*\/ 3 *\/  7 - 4*\/ 3                     
/
∫13sin⁡2(x)+1 dx=C+43(atan⁡(tan⁡(x2)7−43)+π⌊x2−π2π⌋)3037−43+527−43+7(atan⁡(tan⁡(x2)7−43)+π⌊x2−π2π⌋)3037−43+527−43+437−4343+7(atan⁡(tan⁡(x2)43+7)+π⌊x2−π2π⌋)3037−43+527−43+77−4343+7(atan⁡(tan⁡(x2)43+7)+π⌊x2−π2π⌋)3037−43+527−43\int \frac{1}{3 \sin^{2}{\left(x \right)} + 1}\, dx = C + \frac{4 \sqrt{3} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{7 - 4 \sqrt{3}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{30 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 52 \sqrt{7 - 4 \sqrt{3}}} + \frac{7 \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{7 - 4 \sqrt{3}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{30 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 52 \sqrt{7 - 4 \sqrt{3}}} + \frac{4 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} \sqrt{4 \sqrt{3} + 7} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{4 \sqrt{3} + 7}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{30 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 52 \sqrt{7 - 4 \sqrt{3}}} + \frac{7 \sqrt{7 - 4 \sqrt{3}} \sqrt{4 \sqrt{3} + 7} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{4 \sqrt{3} + 7}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{30 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 52 \sqrt{7 - 4 \sqrt{3}}}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.905-5
Trazar el gráfico f(x)
Respuesta [src] 
                                                                                                                                                   _____________    _____________                               
                                                                   /    tan(1/2)    \                                                             /         ___    /         ___  /          /    tan(1/2)    \\
                                                         -pi + atan|----------------|                                                           \/  7 - 4*\/ 3  *\/  7 + 4*\/ 3  *|-pi + atan|----------------||
                                                                   |   _____________|                      _____________    _____________                                         |          |   _____________||
                                                                   |  /         ___ |                     /         ___    /         ___                                          |          |  /         ___ ||
                      pi                                           \\/  7 - 4*\/ 3  /                pi*\/  7 - 4*\/ 3  *\/  7 + 4*\/ 3                                           \          \\/  7 + 4*\/ 3  //
--------------------------------------------- + --------------------------------------------- + --------------------------------------------- + ----------------------------------------------------------------
     _____________              _____________        _____________              _____________        _____________              _____________                 _____________              _____________          
    /         ___        ___   /         ___        /         ___        ___   /         ___        /         ___        ___   /         ___                 /         ___        ___   /         ___           
4*\/  7 - 4*\/ 3   + 2*\/ 3 *\/  7 - 4*\/ 3     4*\/  7 - 4*\/ 3   + 2*\/ 3 *\/  7 - 4*\/ 3     4*\/  7 - 4*\/ 3   + 2*\/ 3 *\/  7 - 4*\/ 3              4*\/  7 - 4*\/ 3   + 2*\/ 3 *\/  7 - 4*\/ 3
7−4343+7(−π+atan⁡(tan⁡(12)43+7))237−43+47−43+−π+atan⁡(tan⁡(12)7−43)237−43+47−43+π7−4343+7237−43+47−43+π237−43+47−43\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{4 \sqrt{3} + 7} \left(- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{4 \sqrt{3} + 7}} \right)}\right)}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}} + \frac{- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{7 - 4 \sqrt{3}}} \right)}}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}} + \frac{\pi \sqrt{7 - 4 \sqrt{3}} \sqrt{4 \sqrt{3} + 7}}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}} + \frac{\pi}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}} 
=
= 
                                                                                                                                                   _____________    _____________                               
                                                                   /    tan(1/2)    \                                                             /         ___    /         ___  /          /    tan(1/2)    \\
                                                         -pi + atan|----------------|                                                           \/  7 - 4*\/ 3  *\/  7 + 4*\/ 3  *|-pi + atan|----------------||
                                                                   |   _____________|                      _____________    _____________                                         |          |   _____________||
                                                                   |  /         ___ |                     /         ___    /         ___                                          |          |  /         ___ ||
                      pi                                           \\/  7 - 4*\/ 3  /                pi*\/  7 - 4*\/ 3  *\/  7 + 4*\/ 3                                           \          \\/  7 + 4*\/ 3  //
--------------------------------------------- + --------------------------------------------- + --------------------------------------------- + ----------------------------------------------------------------
     _____________              _____________        _____________              _____________        _____________              _____________                 _____________              _____________          
    /         ___        ___   /         ___        /         ___        ___   /         ___        /         ___        ___   /         ___                 /         ___        ___   /         ___           
4*\/  7 - 4*\/ 3   + 2*\/ 3 *\/  7 - 4*\/ 3     4*\/  7 - 4*\/ 3   + 2*\/ 3 *\/  7 - 4*\/ 3     4*\/  7 - 4*\/ 3   + 2*\/ 3 *\/  7 - 4*\/ 3              4*\/  7 - 4*\/ 3   + 2*\/ 3 *\/  7 - 4*\/ 3
7−4343+7(−π+atan⁡(tan⁡(12)43+7))237−43+47−43+−π+atan⁡(tan⁡(12)7−43)237−43+47−43+π7−4343+7237−43+47−43+π237−43+47−43\frac{\sqrt{7 - 4 \sqrt{3}} \sqrt{4 \sqrt{3} + 7} \left(- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{4 \sqrt{3} + 7}} \right)}\right)}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}} + \frac{- \pi + \operatorname{atan}{\left(\frac{\tan{\left(\frac{1}{2} \right)}}{\sqrt{7 - 4 \sqrt{3}}} \right)}}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}} + \frac{\pi \sqrt{7 - 4 \sqrt{3}} \sqrt{4 \sqrt{3} + 7}}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}} + \frac{\pi}{2 \sqrt{3} \sqrt{7 - 4 \sqrt{3}} + 4 \sqrt{7 - 4 \sqrt{3}}} 
pi/(4*sqrt(7 - 4*sqrt(3)) + 2*sqrt(3)*sqrt(7 - 4*sqrt(3))) + (-pi + atan(tan(1/2)/sqrt(7 - 4*sqrt(3))))/(4*sqrt(7 - 4*sqrt(3)) + 2*sqrt(3)*sqrt(7 - 4*sqrt(3))) + pi*sqrt(7 - 4*sqrt(3))*sqrt(7 + 4*sqrt(3))/(4*sqrt(7 - 4*sqrt(3)) + 2*sqrt(3)*sqrt(7 - 4*sqrt(3))) + sqrt(7 - 4*sqrt(3))*sqrt(7 + 4*sqrt(3))*(-pi + atan(tan(1/2)/sqrt(7 + 4*sqrt(3))))/(4*sqrt(7 - 4*sqrt(3)) + 2*sqrt(3)*sqrt(7 - 4*sqrt(3)))
Respuesta numérica [src] 
0.630072276228107
0.630072276228107

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

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