Integral of cos2x dx

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

Integral(cos(2*x), (x, 0, 1))

Detail solution

Let $u = 2 x$ .

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

sin(2)
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  2

{{\sin 2}\over{2}}

sin(2)
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  2

\frac{\sin{\left(2 \right)}}{2}

Упростить

Numerical answer [src]

0.454648713412841

0.454648713412841

The graph

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