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Resolución de integrales/ (sinx-1)/ 1/((sinx+4)(sinx-1))

Integral de 1/((sinx+4)(sinx-1)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                             
  /                             
 |                              
 |              1               
 |  ------------------------- dx
 |  (sin(x) + 4)*(sin(x) - 1)   
 |                              
/                               
0
$$\int\limits_{0}^{1} \frac{1}{\left(\sin{\left(x \right)} - 1\right) \left(\sin{\left(x \right)} + 4\right)}\, dx$$ 
Integral(1/((sin(x) + 4)*(sin(x) - 1)), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                                 /        /x   pi\       /             ____    /x\\\            /        /x   pi\       /             ____    /x\\\       
                                                                 |        |- - --|       |  ____   4*\/ 15 *tan|-|||            |        |- - --|       |  ____   4*\/ 15 *tan|-|||       
  /                                                         ____ |        |2   2 |       |\/ 15                \2/||       ____ |        |2   2 |       |\/ 15                \2/||    /x\
 |                                                      2*\/ 15 *|pi*floor|------| + atan|------ + ---------------||   2*\/ 15 *|pi*floor|------| + atan|------ + ---------------||*tan|-|
 |             1                             30                  \        \  pi  /       \  15            15      //            \        \  pi  /       \  15            15      //    \2/
 | ------------------------- dx = C + --------------- + ------------------------------------------------------------ - -------------------------------------------------------------------
 | (sin(x) + 4)*(sin(x) - 1)                      /x\                                     /x\                                                                /x\                          
 |                                    -75 + 75*tan|-|                         -75 + 75*tan|-|                                                    -75 + 75*tan|-|                          
/                                                 \2/                                     \2/                                                                \2/
$$\int \frac{1}{\left(\sin{\left(x \right)} - 1\right) \left(\sin{\left(x \right)} + 4\right)}\, dx = C - \frac{2 \sqrt{15} \left(\operatorname{atan}{\left(\frac{4 \sqrt{15} \tan{\left(\frac{x}{2} \right)}}{15} + \frac{\sqrt{15}}{15} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right) \tan{\left(\frac{x}{2} \right)}}{75 \tan{\left(\frac{x}{2} \right)} - 75} + \frac{2 \sqrt{15} \left(\operatorname{atan}{\left(\frac{4 \sqrt{15} \tan{\left(\frac{x}{2} \right)}}{15} + \frac{\sqrt{15}}{15} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{75 \tan{\left(\frac{x}{2} \right)} - 75} + \frac{30}{75 \tan{\left(\frac{x}{2} \right)} - 75}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                 /          /  ____\\            /          /  ____       ____         \\            /          /  ____       ____         \\         
                            ____ |          |\/ 15 ||       ____ |          |\/ 15    4*\/ 15 *tan(1/2)||       ____ |          |\/ 15    4*\/ 15 *tan(1/2)||         
                        2*\/ 15 *|-pi + atan|------||   2*\/ 15 *|-pi + atan|------ + -----------------||   2*\/ 15 *|-pi + atan|------ + -----------------||*tan(1/2)
2           30                   \          \  15  //            \          \  15             15       //            \          \  15             15       //         
- + ----------------- + ----------------------------- + ------------------------------------------------- - ----------------------------------------------------------
5   -75 + 75*tan(1/2)                 75                                -75 + 75*tan(1/2)                                       -75 + 75*tan(1/2)
$$\frac{30}{-75 + 75 \tan{\left(\frac{1}{2} \right)}} - \frac{2 \sqrt{15} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{15}}{15} + \frac{4 \sqrt{15} \tan{\left(\frac{1}{2} \right)}}{15} \right)}\right) \tan{\left(\frac{1}{2} \right)}}{-75 + 75 \tan{\left(\frac{1}{2} \right)}} + \frac{2 \sqrt{15} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)}\right)}{75} + \frac{2}{5} + \frac{2 \sqrt{15} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{15}}{15} + \frac{4 \sqrt{15} \tan{\left(\frac{1}{2} \right)}}{15} \right)}\right)}{-75 + 75 \tan{\left(\frac{1}{2} \right)}}$$ 
=
= 
                                 /          /  ____\\            /          /  ____       ____         \\            /          /  ____       ____         \\         
                            ____ |          |\/ 15 ||       ____ |          |\/ 15    4*\/ 15 *tan(1/2)||       ____ |          |\/ 15    4*\/ 15 *tan(1/2)||         
                        2*\/ 15 *|-pi + atan|------||   2*\/ 15 *|-pi + atan|------ + -----------------||   2*\/ 15 *|-pi + atan|------ + -----------------||*tan(1/2)
2           30                   \          \  15  //            \          \  15             15       //            \          \  15             15       //         
- + ----------------- + ----------------------------- + ------------------------------------------------- - ----------------------------------------------------------
5   -75 + 75*tan(1/2)                 75                                -75 + 75*tan(1/2)                                       -75 + 75*tan(1/2)
$$\frac{30}{-75 + 75 \tan{\left(\frac{1}{2} \right)}} - \frac{2 \sqrt{15} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{15}}{15} + \frac{4 \sqrt{15} \tan{\left(\frac{1}{2} \right)}}{15} \right)}\right) \tan{\left(\frac{1}{2} \right)}}{-75 + 75 \tan{\left(\frac{1}{2} \right)}} + \frac{2 \sqrt{15} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)}\right)}{75} + \frac{2}{5} + \frac{2 \sqrt{15} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{15}}{15} + \frac{4 \sqrt{15} \tan{\left(\frac{1}{2} \right)}}{15} \right)}\right)}{-75 + 75 \tan{\left(\frac{1}{2} \right)}}$$ 
2/5 + 30/(-75 + 75*tan(1/2)) + 2*sqrt(15)*(-pi + atan(sqrt(15)/15))/75 + 2*sqrt(15)*(-pi + atan(sqrt(15)/15 + 4*sqrt(15)*tan(1/2)/15))/(-75 + 75*tan(1/2)) - 2*sqrt(15)*(-pi + atan(sqrt(15)/15 + 4*sqrt(15)*tan(1/2)/15))*tan(1/2)/(-75 + 75*tan(1/2))
Respuesta numérica [src] 
-0.526631333141361
-0.526631333141361

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

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