Integral of sin(3pix) dx

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  1               
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 |  sin(3*pi*x) dx
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$$\int\limits_{-1}^{1} \sin{\left(3 \pi x \right)}\, dx$$

Integral(sin((3*pi)*x), (x, -1, 1))

Detail solution

Let $u = 3 \pi x$ .

Then let $du = 3 \pi dx$ and substitute $\frac{du}{3 \pi}$ :

$\int \frac{\sin{\left(u \right)}}{3 \pi}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{3 \pi}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- \frac{\cos{\left(u \right)}}{3 \pi}$
Now substitute $u$ back in:

$- \frac{\cos{\left(3 \pi x \right)}}{3 \pi}$
Now simplify:

$- \frac{\cos{\left(3 \pi x \right)}}{3 \pi}$
Add the constant of integration:

$- \frac{\cos{\left(3 \pi x \right)}}{3 \pi}+ \mathrm{constant}$

The answer is:

$- \frac{\cos{\left(3 \pi x \right)}}{3 \pi}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                
 |                      cos(3*pi*x)
 | sin(3*pi*x) dx = C - -----------
 |                          3*pi   
/

$$\int \sin{\left(3 \pi x \right)}\, dx = C - \frac{\cos{\left(3 \pi x \right)}}{3 \pi}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Numerical answer [src]

0.0

0.0

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

Integral of sin(3*pi*x) dx