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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |       2        
 |    sin (x)     
 |  ----------- dx
 |         2      
 |  1 + cos (x)   
 |                
/                 
0
∫01sin⁡2(x)cos⁡2(x)+1 dx\int\limits_{0}^{1} \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} + 1}\, dx 
Integral(sin(x)^2/(1 + cos(x)^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                                                               
 |                                /        /x   pi\                         \         /        /x   pi\                          \
 |      2                         |        |- - --|                         |         |        |- - --|                          |
 |   sin (x)                  ___ |        |2   2 |       /      ___    /x\\|     ___ |        |2   2 |       /       ___    /x\\|
 | ----------- dx = C - x + \/ 2 *|pi*floor|------| + atan|1 + \/ 2 *tan|-||| + \/ 2 *|pi*floor|------| + atan|-1 + \/ 2 *tan|-|||
 |        2                       \        \  pi  /       \             \2///         \        \  pi  /       \              \2///
 | 1 + cos (x)                                                                                                                    
 |                                                                                                                                
/
∫sin⁡2(x)cos⁡2(x)+1 dx=C−x+2(atan⁡(2tan⁡(x2)−1)+π⌊x2−π2π⌋)+2(atan⁡(2tan⁡(x2)+1)+π⌊x2−π2π⌋)\int \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} + 1}\, dx = C - x + \sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right) + \sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90-1010
Trazar el gráfico f(x)
Respuesta [src] 
       ___ /          /      ___         \\     ___ /          /      ___         \\          ___
-1 + \/ 2 *\-pi - atan\1 - \/ 2 *tan(1/2)// + \/ 2 *\-pi + atan\1 + \/ 2 *tan(1/2)// + 2*pi*\/ 2
2(−π−atan⁡(−2tan⁡(12)+1))+2(−π+atan⁡(2tan⁡(12)+1))−1+22π\sqrt{2} \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right) + \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right) - 1 + 2 \sqrt{2} \pi 
=
= 
       ___ /          /      ___         \\     ___ /          /      ___         \\          ___
-1 + \/ 2 *\-pi - atan\1 - \/ 2 *tan(1/2)// + \/ 2 *\-pi + atan\1 + \/ 2 *tan(1/2)// + 2*pi*\/ 2
2(−π−atan⁡(−2tan⁡(12)+1))+2(−π+atan⁡(2tan⁡(12)+1))−1+22π\sqrt{2} \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right) + \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right) - 1 + 2 \sqrt{2} \pi 
-1 + sqrt(2)*(-pi - atan(1 - sqrt(2)*tan(1/2))) + sqrt(2)*(-pi + atan(1 + sqrt(2)*tan(1/2))) + 2*pi*sqrt(2)
Respuesta numérica [src] 
0.178815078927437
0.178815078927437

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

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