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Solving integrals/ sin(pi*x/(2*y))

Integral of sin(pi*x/(2*y)) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  y             
  /             
 |              
 |     /pi*x\   
 |  sin|----| dx
 |     \2*y /   
 |              
/               
 2              
y
$$\int\limits_{y^{2}}^{y} \sin{\left(\frac{\pi x}{2 y} \right)}\, dx$$ 
Integral(sin((pi*x)/((2*y))), (x, y^2, y))
Detail solution

  1. Let .

    Then let and substitute :

    1. The integral of a constant times a function is the constant times the integral of the function:

      1. The integral of sine is negative cosine:

      So, the result is:

    Now substitute back in:

  2. Now simplify:

  3. Add the constant of integration:

The answer is:

The answer (Indefinite) [src] 
  /                          /pi*x\
 |                    2*y*cos|----|
 |    /pi*x\                 \2*y /
 | sin|----| dx = C - -------------
 |    \2*y /                pi     
 |                                 
/
$$\int \sin{\left(\frac{\pi x}{2 y} \right)}\, dx = C - \frac{2 y \cos{\left(\frac{\pi x}{2 y} \right)}}{\pi}$$
Simplify
The answer [src] 
/       /pi*y\                                  
|2*y*cos|----|                                  
|       \ 2  /                                  
<-------------  for And(y > -oo, y < oo, y != 0)
|      pi                                       
|                                               
\      0                   otherwise
$$\begin{cases} \frac{2 y \cos{\left(\frac{\pi y}{2} \right)}}{\pi} & \text{for}\: y > -\infty \wedge y < \infty \wedge y \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
=
= 
/       /pi*y\                                  
|2*y*cos|----|                                  
|       \ 2  /                                  
<-------------  for And(y > -oo, y < oo, y != 0)
|      pi                                       
|                                               
\      0                   otherwise
$$\begin{cases} \frac{2 y \cos{\left(\frac{\pi y}{2} \right)}}{\pi} & \text{for}\: y > -\infty \wedge y < \infty \wedge y \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
Piecewise((2*y*cos(pi*y/2)/pi, (y > -oo)∧(y < oo)∧(Ne(y, 0))), (0, True))

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

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