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

Integral of sin(pix/(2y)) dx

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

Let $u = \frac{\pi x}{2 y}$ .

Then let $du = \frac{\pi dx}{2 y}$ and substitute $\frac{2 du y}{\pi}$ :

$\int \frac{2 y \sin{\left(u \right)}}{\pi}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sin{\left(u \right)}\, du = \frac{2 y \int \sin{\left(u \right)}\, du}{\pi}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- \frac{2 y \cos{\left(u \right)}}{\pi}$
Now substitute $u$ back in:

$- \frac{2 y \cos{\left(\frac{\pi x}{2 y} \right)}}{\pi}$
Now simplify:

$- \frac{2 y \cos{\left(\frac{\pi x}{2 y} \right)}}{\pi}$
Add the constant of integration:

$- \frac{2 y \cos{\left(\frac{\pi x}{2 y} \right)}}{\pi}+ \mathrm{constant}$

The answer is:

$- \frac{2 y \cos{\left(\frac{\pi x}{2 y} \right)}}{\pi}+ \mathrm{constant}$

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.

Other calculators

How to use it?

Identical expressions

Integral of sin(pix/(2y)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Integral of sin(pix/(2y)) dx