Integral of sin(pi*x/2) dx

The solution

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  1             
  /             
 |              
 |     /pi*x\   
 |  sin|----| dx
 |     \ 2  /   
 |              
/               
0

$$\int\limits_{0}^{1} \sin{\left(\frac{\pi x}{2} \right)}\, dx$$

Integral(sin((pi*x)/2), (x, 0, 1))

Detail solution

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

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

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                        /pi*x\
 |                    2*cos|----|
 |    /pi*x\               \ 2  /
 | sin|----| dx = C - -----------
 |    \ 2  /               pi    
 |                               
/

$$\int \sin{\left(\frac{\pi x}{2} \right)}\, dx = C - \frac{2 \cos{\left(\frac{\pi x}{2} \right)}}{\pi}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

2 
--
pi

$$\frac{2}{\pi}$$

2 
--
pi

$$\frac{2}{\pi}$$

2/pi

Numerical answer [src]

0.636619772367581

0.636619772367581

The graph

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

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Integral of sin(pi*x/2) dx

Limits of integration:

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The solution