Integral of cos(2pi(x)) dx

You have entered [src]

  1               
  /               
 |                
 |  cos(2*pi*x) dx
 |                
/                 
0

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

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   
/

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

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.