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Resolución de integrales/ sin(x)/ sin(x)*dx/(5+4*sin(x))

Integral de sin(x)*dx/(5+4*sin(x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  p                
  -                
  2                
  /                
 |                 
 |     sin(x)      
 |  ------------ dx
 |  5 + 4*sin(x)   
 |                 
/                  
0
∫0p2sin⁡(x)4sin⁡(x)+5 dx\int\limits_{0}^{\frac{p}{2}} \frac{\sin{\left(x \right)}}{4 \sin{\left(x \right)} + 5}\, dx 
Integral(sin(x)/(5 + 4*sin(x)), (x, 0, p/2))
Respuesta (Indefinida) [src] 
                               /         /x\\                 /x   pi\
                               |    5*tan|-||                 |- - --|
  /                            |4        \2/|                 |2   2 |
 |                       5*atan|- + --------|       5*pi*floor|------|
 |    sin(x)                   \3      3    /   x             \  pi  /
 | ------------ dx = C - -------------------- + - - ------------------
 | 5 + 4*sin(x)                   6             4           6         
 |                                                                    
/
∫sin⁡(x)4sin⁡(x)+5 dx=C+x4−5atan⁡(5tan⁡(x2)3+43)6−5π⌊x2−π2π⌋6\int \frac{\sin{\left(x \right)}}{4 \sin{\left(x \right)} + 5}\, dx = C + \frac{x}{4} - \frac{5 \operatorname{atan}{\left(\frac{5 \tan{\left(\frac{x}{2} \right)}}{3} + \frac{4}{3} \right)}}{6} - \frac{5 \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor}{6}
Simplificar
Respuesta [src] 
               /         /p\\                               /  pi   p\
               |    5*tan|-||                               |- -- + -|
               |4        \4/|                               |  2    4|
         5*atan|- + --------|                     5*pi*floor|--------|
  5*pi         \3      3    /   p   5*atan(4/3)             \   pi   /
- ---- - -------------------- + - + ----------- - --------------------
   6              6             8        6                 6
p8−5atan⁡(5tan⁡(p4)3+43)6−5π⌊p4−π2π⌋6−5π6+5atan⁡(43)6\frac{p}{8} - \frac{5 \operatorname{atan}{\left(\frac{5 \tan{\left(\frac{p}{4} \right)}}{3} + \frac{4}{3} \right)}}{6} - \frac{5 \pi \left\lfloor{\frac{\frac{p}{4} - \frac{\pi}{2}}{\pi}}\right\rfloor}{6} - \frac{5 \pi}{6} + \frac{5 \operatorname{atan}{\left(\frac{4}{3} \right)}}{6} 
=
= 
               /         /p\\                               /  pi   p\
               |    5*tan|-||                               |- -- + -|
               |4        \4/|                               |  2    4|
         5*atan|- + --------|                     5*pi*floor|--------|
  5*pi         \3      3    /   p   5*atan(4/3)             \   pi   /
- ---- - -------------------- + - + ----------- - --------------------
   6              6             8        6                 6
p8−5atan⁡(5tan⁡(p4)3+43)6−5π⌊p4−π2π⌋6−5π6+5atan⁡(43)6\frac{p}{8} - \frac{5 \operatorname{atan}{\left(\frac{5 \tan{\left(\frac{p}{4} \right)}}{3} + \frac{4}{3} \right)}}{6} - \frac{5 \pi \left\lfloor{\frac{\frac{p}{4} - \frac{\pi}{2}}{\pi}}\right\rfloor}{6} - \frac{5 \pi}{6} + \frac{5 \operatorname{atan}{\left(\frac{4}{3} \right)}}{6} 
-5*pi/6 - 5*atan(4/3 + 5*tan(p/4)/3)/6 + p/8 + 5*atan(4/3)/6 - 5*pi*floor((-pi/2 + p/4)/pi)/6

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

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