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Solving integrals/ x*sinx*cos(2x/pi)

Integral of x*sinx*cos(2x/pi) dx

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The solution

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 pi                     
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 |              /2*x\   
 |  x*sin(x)*cos|---| dx
 |              \ pi/   
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0
∫0πxsin⁡(x)cos⁡(2xπ) dx\int\limits_{0}^{\pi} x \sin{\left(x \right)} \cos{\left(\frac{2 x}{\pi} \right)}\, dx 
Integral((x*sin(x))*cos((2*x)/pi), (x, 0, pi))
The answer (Indefinite) [src] 
  /                               4    /    2*x\        4    /    2*x\         3    /    2*x\        2    /    2*x\        2    /    2*x\        3    /    2*x\        4    /    2*x\        4    /    2*x\             /    2*x\         3    /    2*x\         3    /    2*x\         2    /    2*x\         2    /    2*x\             /    2*x\
 |                              pi *sin|x - ---|      pi *sin|x + ---|     4*pi *sin|x + ---|    4*pi *sin|x - ---|    4*pi *sin|x + ---|    4*pi *sin|x - ---|    x*pi *cos|x - ---|    x*pi *cos|x + ---|   8*pi*x*cos|x + ---|   2*x*pi *cos|x - ---|   2*x*pi *cos|x + ---|   4*x*pi *cos|x - ---|   4*x*pi *cos|x + ---|   8*pi*x*cos|x - ---|
 |             /2*x\                   \     pi/             \     pi/              \     pi/             \     pi/             \     pi/             \     pi/             \     pi/             \     pi/             \     pi/              \     pi/              \     pi/              \     pi/              \     pi/             \     pi/
 | x*sin(x)*cos|---| dx = C + ------------------- + ------------------- - ------------------- + ------------------- + ------------------- + ------------------- - ------------------- - ------------------- - ------------------- - -------------------- + -------------------- + -------------------- + -------------------- + -------------------
 |             \ pi/                    2       4             2       4             2       4             2       4             2       4             2       4             2       4             2       4             2       4             2       4              2       4              2       4              2       4              2       4
 |                            32 - 16*pi  + 2*pi    32 - 16*pi  + 2*pi    32 - 16*pi  + 2*pi    32 - 16*pi  + 2*pi    32 - 16*pi  + 2*pi    32 - 16*pi  + 2*pi    32 - 16*pi  + 2*pi    32 - 16*pi  + 2*pi    32 - 16*pi  + 2*pi    32 - 16*pi  + 2*pi     32 - 16*pi  + 2*pi     32 - 16*pi  + 2*pi     32 - 16*pi  + 2*pi     32 - 16*pi  + 2*pi 
/
∫xsin⁡(x)cos⁡(2xπ) dx=C−π4xcos⁡(−2xπ+x)−16π2+32+2π4−2π3xcos⁡(−2xπ+x)−16π2+32+2π4+8πxcos⁡(−2xπ+x)−16π2+32+2π4+4π2xcos⁡(−2xπ+x)−16π2+32+2π4−π4xcos⁡(2xπ+x)−16π2+32+2π4−8πxcos⁡(2xπ+x)−16π2+32+2π4+4π2xcos⁡(2xπ+x)−16π2+32+2π4+2π3xcos⁡(2xπ+x)−16π2+32+2π4+4π2sin⁡(−2xπ+x)−16π2+32+2π4+π4sin⁡(−2xπ+x)−16π2+32+2π4+4π3sin⁡(−2xπ+x)−16π2+32+2π4−4π3sin⁡(2xπ+x)−16π2+32+2π4+4π2sin⁡(2xπ+x)−16π2+32+2π4+π4sin⁡(2xπ+x)−16π2+32+2π4\int x \sin{\left(x \right)} \cos{\left(\frac{2 x}{\pi} \right)}\, dx = C - \frac{\pi^{4} x \cos{\left(- \frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} - \frac{2 \pi^{3} x \cos{\left(- \frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{8 \pi x \cos{\left(- \frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{4 \pi^{2} x \cos{\left(- \frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} - \frac{\pi^{4} x \cos{\left(\frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} - \frac{8 \pi x \cos{\left(\frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{4 \pi^{2} x \cos{\left(\frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{2 \pi^{3} x \cos{\left(\frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{4 \pi^{2} \sin{\left(- \frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{\pi^{4} \sin{\left(- \frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{4 \pi^{3} \sin{\left(- \frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} - \frac{4 \pi^{3} \sin{\left(\frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{4 \pi^{2} \sin{\left(\frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}} + \frac{\pi^{4} \sin{\left(\frac{2 x}{\pi} + x \right)}}{- 16 \pi^{2} + 32 + 2 \pi^{4}}
Simplify
The graph
0.000.250.500.751.001.251.501.752.002.252.502.753.002-1
Plot the graph f(x)
The answer [src] 
     5                   3                  3         
   pi *cos(2)        4*pi *cos(2)       4*pi *sin(2)  
---------------- - ---------------- + ----------------
       4       2          4       2          4       2
16 + pi  - 8*pi    16 + pi  - 8*pi    16 + pi  - 8*pi
π5cos⁡(2)−8π2+16+π4−4π3cos⁡(2)−8π2+16+π4+4π3sin⁡(2)−8π2+16+π4\frac{\pi^{5} \cos{\left(2 \right)}}{- 8 \pi^{2} + 16 + \pi^{4}} - \frac{4 \pi^{3} \cos{\left(2 \right)}}{- 8 \pi^{2} + 16 + \pi^{4}} + \frac{4 \pi^{3} \sin{\left(2 \right)}}{- 8 \pi^{2} + 16 + \pi^{4}} 
=
= 
     5                   3                  3         
   pi *cos(2)        4*pi *cos(2)       4*pi *sin(2)  
---------------- - ---------------- + ----------------
       4       2          4       2          4       2
16 + pi  - 8*pi    16 + pi  - 8*pi    16 + pi  - 8*pi
π5cos⁡(2)−8π2+16+π4−4π3cos⁡(2)−8π2+16+π4+4π3sin⁡(2)−8π2+16+π4\frac{\pi^{5} \cos{\left(2 \right)}}{- 8 \pi^{2} + 16 + \pi^{4}} - \frac{4 \pi^{3} \cos{\left(2 \right)}}{- 8 \pi^{2} + 16 + \pi^{4}} + \frac{4 \pi^{3} \sin{\left(2 \right)}}{- 8 \pi^{2} + 16 + \pi^{4}} 
pi^5*cos(2)/(16 + pi^4 - 8*pi^2) - 4*pi^3*cos(2)/(16 + pi^4 - 8*pi^2) + 4*pi^3*sin(2)/(16 + pi^4 - 8*pi^2)
Numerical answer [src] 
1.0750890351302
1.0750890351302

    Use the examples entering the upper and lower limits of integration.

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