Mister Exam

# Integral of (sin(2021)x)/(pi*sin(x)) dx

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### The solution

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 pi
/
|
|  sin(2021)*x
|  ----------- dx
|   pi*sin(x)
|
/
0                 
$$\int\limits_{0}^{\pi} \frac{x \sin{\left(2021 \right)}}{\pi \sin{\left(x \right)}}\, dx$$
Integral(sin(2021)*x/((pi*sin(x))), (x, 0, pi))
Detail solution
1. The integral of a constant times a function is the constant times the integral of the function:

1. Don't know the steps in finding this integral.

But the integral is

So, the result is:

2. Add the constant of integration:

                          /
|
|   x
| ------ dx*sin(2021)
/                      | sin(x)
|                       |
| sin(2021)*x          /
| ----------- dx = C + ----------------------
|  pi*sin(x)                     pi
|
/                                             
$$-{{\sin 2021\,\left(x\,\log \left(\sin ^2x+\cos ^2x+2\,\cos x+1 \right)-x\,\log \left(\sin ^2x+\cos ^2x-2\,\cos x+1\right)+2\,i\,x\, {\rm atan2}\left(\sin x , \cos x+1\right)+2\,i\,x\,{\rm atan2}\left( \sin x , 1-\cos x\right)+2\,i\,{\it li}_{2}(e^{i\,x})-2\,i\,{\it li} _{2}(-e^{i\,x})\right)}\over{2\,\pi}}$$
 pi
/
|
|    x
|  ------ dx*sin(2021)
|  sin(x)
|
/
0
-----------------------
pi          
$$\frac{\sin{\left(2021 \right)} \int\limits_{0}^{\pi} \frac{x}{\sin{\left(x \right)}}\, dx}{\pi}$$
=
=
 pi
/
|
|    x
|  ------ dx*sin(2021)
|  sin(x)
|
/
0
-----------------------
pi          
$$\frac{\sin{\left(2021 \right)} \int\limits_{0}^{\pi} \frac{x}{\sin{\left(x \right)}}\, dx}{\pi}$$
-30.4929541152786
-30.4929541152786