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Resolución de integrales/ x*sinx/ x*sinx*cos(x/pi)

Integral de x*sinx*cos(x/pi) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                    
  /                    
 |                     
 |              /x \   
 |  x*sin(x)*cos|--| dx
 |              \pi/   
 |                     
/                      
0
∫0πxsin⁡(x)cos⁡(xπ) dx\int\limits_{0}^{\pi} x \sin{\left(x \right)} \cos{\left(\frac{x}{\pi} \right)}\, dx 
Integral((x*sin(x))*cos(x/pi), (x, 0, pi))
Respuesta (Indefinida) [src] 
  /                            2    /x \            4    /x \              3           /x \                  /x \       2           /x \       3           /x \       4           /x \
 |                           pi *cos|--|*sin(x)   pi *cos|--|*sin(x)   2*pi *cos(x)*sin|--|   pi*x*sin(x)*sin|--|   x*pi *cos(x)*cos|--|   x*pi *sin(x)*sin|--|   x*pi *cos(x)*cos|--|
 |             /x \                 \pi/                 \pi/                          \pi/                  \pi/                   \pi/                   \pi/                   \pi/
 | x*sin(x)*cos|--| dx = C + ------------------ + ------------------ - -------------------- + ------------------- + -------------------- - -------------------- - --------------------
 |             \pi/                 4       2            4       2             4       2              4       2             4       2              4       2              4       2   
 |                            1 + pi  - 2*pi       1 + pi  - 2*pi        1 + pi  - 2*pi         1 + pi  - 2*pi        1 + pi  - 2*pi         1 + pi  - 2*pi         1 + pi  - 2*pi    
/
∫xsin⁡(x)cos⁡(xπ) dx=C−π3xsin⁡(x)sin⁡(xπ)−2π2+1+π4+πxsin⁡(x)sin⁡(xπ)−2π2+1+π4−π4xcos⁡(x)cos⁡(xπ)−2π2+1+π4+π2xcos⁡(x)cos⁡(xπ)−2π2+1+π4+π2sin⁡(x)cos⁡(xπ)−2π2+1+π4+π4sin⁡(x)cos⁡(xπ)−2π2+1+π4−2π3sin⁡(xπ)cos⁡(x)−2π2+1+π4\int x \sin{\left(x \right)} \cos{\left(\frac{x}{\pi} \right)}\, dx = C - \frac{\pi^{3} x \sin{\left(x \right)} \sin{\left(\frac{x}{\pi} \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{\pi x \sin{\left(x \right)} \sin{\left(\frac{x}{\pi} \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} - \frac{\pi^{4} x \cos{\left(x \right)} \cos{\left(\frac{x}{\pi} \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{\pi^{2} x \cos{\left(x \right)} \cos{\left(\frac{x}{\pi} \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{\pi^{2} \sin{\left(x \right)} \cos{\left(\frac{x}{\pi} \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{\pi^{4} \sin{\left(x \right)} \cos{\left(\frac{x}{\pi} \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} - \frac{2 \pi^{3} \sin{\left(\frac{x}{\pi} \right)} \cos{\left(x \right)}}{- 2 \pi^{2} + 1 + \pi^{4}}
Simplificar
Gráfica
0.000.250.500.751.001.251.501.752.002.252.502.753.000.05.0
Trazar el gráfico f(x)
Respuesta [src] 
     5                 3                  3        
   pi *cos(1)        pi *cos(1)       2*pi *sin(1) 
--------------- - --------------- + ---------------
      4       2         4       2         4       2
1 + pi  - 2*pi    1 + pi  - 2*pi    1 + pi  - 2*pi
−π3cos⁡(1)−2π2+1+π4+2π3sin⁡(1)−2π2+1+π4+π5cos⁡(1)−2π2+1+π4- \frac{\pi^{3} \cos{\left(1 \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{2 \pi^{3} \sin{\left(1 \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{\pi^{5} \cos{\left(1 \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} 
=
= 
     5                 3                  3        
   pi *cos(1)        pi *cos(1)       2*pi *sin(1) 
--------------- - --------------- + ---------------
      4       2         4       2         4       2
1 + pi  - 2*pi    1 + pi  - 2*pi    1 + pi  - 2*pi
−π3cos⁡(1)−2π2+1+π4+2π3sin⁡(1)−2π2+1+π4+π5cos⁡(1)−2π2+1+π4- \frac{\pi^{3} \cos{\left(1 \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{2 \pi^{3} \sin{\left(1 \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} + \frac{\pi^{5} \cos{\left(1 \right)}}{- 2 \pi^{2} + 1 + \pi^{4}} 
pi^5*cos(1)/(1 + pi^4 - 2*pi^2) - pi^3*cos(1)/(1 + pi^4 - 2*pi^2) + 2*pi^3*sin(1)/(1 + pi^4 - 2*pi^2)
Respuesta numérica [src] 
2.55208393853876
2.55208393853876

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

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