Integral of sin(pix) dx

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  1             
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 |  sin(pi*x) dx
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0

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

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

Detail solution

Let $u = \pi x$ .

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                            
 |                    cos(pi*x)
 | sin(pi*x) dx = C - ---------
 |                        pi   
/

$$-{{\cos \left(\pi\,x\right)}\over{\pi}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

2 
--
pi

$${{1}\over{\pi}}-{{\cos \pi}\over{\pi}}$$

2 
--
pi

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

Simplify

Numerical answer [src]

0.636619772367581

0.636619772367581

The graph

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