Integral of cos(2pi(x)) dx

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 |  cos(2*pi*x) dx
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\int\limits_{0}^{1} \cos{\left(2 \pi x \right)}\, dx

Simplify

Detail solution

Let $u = 2 \pi x$ .

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

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

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

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

The answer is:

\frac{\sin{\left(2 \pi x \right)}}{2 \pi}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                                
 |                      sin(2*pi*x)
 | cos(2*pi*x) dx = C + -----------
 |                          2*pi   
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{{\sin \left(2\,\pi\,x\right)}\over{2\,\pi}}

Simplify

The graph

Plot the graph f(x)

The answer [src]

{{\sin \left(2\,\pi\right)}\over{2\,\pi}}

0

Simplify

Numerical answer [src]

-7.41845798679675e-20

-7.41845798679675e-20

The graph

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