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Resolución de integrales/ sin2(x)/ cos^2/ sin2(x)/(4+cos^2(x))

Integral de sin2(x)/(4+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      
 |  4 + cos (x)   
 |                
/                 
0
∫01sin⁡2(x)cos⁡2(x)+4 dx\int\limits_{0}^{1} \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} + 4}\, dx 
Integral(sin(x)^2/(4 + cos(x)^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                  /        /x   pi\       /      ___    /x\\\         /        /x   pi\       /        ___    /x\\\
  /                               |        |- - --|       |    \/ 5 *tan|-|||         |        |- - --|       |      \/ 5 *tan|-|||
 |                            ___ |        |2   2 |       |1            \2/||     ___ |        |2   2 |       |  1            \2/||
 |      2                   \/ 5 *|pi*floor|------| + atan|- + ------------||   \/ 5 *|pi*floor|------| + atan|- - + ------------||
 |   sin (x)                      \        \  pi  /       \2        2      //         \        \  pi  /       \  2        2      //
 | ----------- dx = C - x + ------------------------------------------------- + ---------------------------------------------------
 |        2                                         2                                                    2                         
 | 4 + cos (x)                                                                                                                     
 |                                                                                                                                 
/
∫sin⁡2(x)cos⁡2(x)+4 dx=C−x+5(atan⁡(5tan⁡(x2)2−12)+π⌊x2−π2π⌋)2+5(atan⁡(5tan⁡(x2)2+12)+π⌊x2−π2π⌋)2\int \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} + 4}\, dx = C - x + \frac{\sqrt{5} \left(\operatorname{atan}{\left(\frac{\sqrt{5} \tan{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2} + \frac{\sqrt{5} \left(\operatorname{atan}{\left(\frac{\sqrt{5} \tan{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90-1010
Trazar el gráfico f(x)
Respuesta [src] 
           /          /      ___         \\         /          /      ___         \\                                                    
       ___ |          |1   \/ 5 *tan(1/2)||     ___ |          |1   \/ 5 *tan(1/2)||                                                    
     \/ 5 *|-pi - atan|- - --------------||   \/ 5 *|-pi + atan|- + --------------||     ___                       ___                  
           \          \2         2       //         \          \2         2       //   \/ 5 *(-pi - atan(1/2))   \/ 5 *(-pi + atan(1/2))
-1 + -------------------------------------- + -------------------------------------- - ----------------------- - -----------------------
                       2                                        2                                 2                         2
5(−π−atan⁡(−5tan⁡(12)2+12))2+5(−π+atan⁡(12+5tan⁡(12)2))2−1−5(−π+atan⁡(12))2−5(−π−atan⁡(12))2\frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(- \frac{\sqrt{5} \tan{\left(\frac{1}{2} \right)}}{2} + \frac{1}{2} \right)}\right)}{2} + \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5} \tan{\left(\frac{1}{2} \right)}}{2} \right)}\right)}{2} - 1 - \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(\frac{1}{2} \right)}\right)}{2} - \frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(\frac{1}{2} \right)}\right)}{2} 
=
= 
           /          /      ___         \\         /          /      ___         \\                                                    
       ___ |          |1   \/ 5 *tan(1/2)||     ___ |          |1   \/ 5 *tan(1/2)||                                                    
     \/ 5 *|-pi - atan|- - --------------||   \/ 5 *|-pi + atan|- + --------------||     ___                       ___                  
           \          \2         2       //         \          \2         2       //   \/ 5 *(-pi - atan(1/2))   \/ 5 *(-pi + atan(1/2))
-1 + -------------------------------------- + -------------------------------------- - ----------------------- - -----------------------
                       2                                        2                                 2                         2
5(−π−atan⁡(−5tan⁡(12)2+12))2+5(−π+atan⁡(12+5tan⁡(12)2))2−1−5(−π+atan⁡(12))2−5(−π−atan⁡(12))2\frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(- \frac{\sqrt{5} \tan{\left(\frac{1}{2} \right)}}{2} + \frac{1}{2} \right)}\right)}{2} + \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5} \tan{\left(\frac{1}{2} \right)}}{2} \right)}\right)}{2} - 1 - \frac{\sqrt{5} \left(- \pi + \operatorname{atan}{\left(\frac{1}{2} \right)}\right)}{2} - \frac{\sqrt{5} \left(- \pi - \operatorname{atan}{\left(\frac{1}{2} \right)}\right)}{2} 
-1 + sqrt(5)*(-pi - atan(1/2 - sqrt(5)*tan(1/2)/2))/2 + sqrt(5)*(-pi + atan(1/2 + sqrt(5)*tan(1/2)/2))/2 - sqrt(5)*(-pi - atan(1/2))/2 - sqrt(5)*(-pi + atan(1/2))/2
Respuesta numérica [src] 
0.060086266205764
0.060086266205764

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

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