Integral of sin(4pix) dx

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  1               
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 |  sin(4*pi*x) dx
 |                
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0

$$\int\limits_{0}^{1} \sin{\left(4 \pi x \right)}\, dx$$

Simplify

Detail solution

Let $u = 4 \pi x$ .

Then let $du = 4 \pi dx$ and substitute $\frac{du}{4 \pi}$ :

$\int \frac{\sin{\left(u \right)}}{16 \pi^{2}}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\sin{\left(u \right)}}{4 \pi}\, du = \frac{\int \sin{\left(u \right)}\, du}{4 \pi}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- \frac{\cos{\left(u \right)}}{4 \pi}$
Now substitute $u$ back in:

$- \frac{\cos{\left(4 \pi x \right)}}{4 \pi}$
Add the constant of integration:

$- \frac{\cos{\left(4 \pi x \right)}}{4 \pi}+ \mathrm{constant}$

The answer is:

$- \frac{\cos{\left(4 \pi x \right)}}{4 \pi}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                
 |                      cos(4*pi*x)
 | sin(4*pi*x) dx = C - -----------
 |                          4*pi   
/

$$-{{\cos \left(4\,\pi\,x\right)}\over{4\,\pi}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$${{1}\over{4\,\pi}}-{{\cos \left(4\,\pi\right)}\over{4\,\pi}}$$

$$0$$

Simplify

Numerical answer [src]

-3.07093340903824e-23

-3.07093340903824e-23

The graph

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