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cos^5(2x)

Integral of cos^5(2x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1             
  /             
 |              
 |     5        
 |  cos (2*x) dx
 |              
/               
0               
$$\int\limits_{0}^{1} \cos^{5}{\left(2 x \right)}\, dx$$
Detail solution
  1. Rewrite the integrand:

  2. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      1. Integrate term-by-term:

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. Let .

            Then let and substitute :

            1. The integral of is when :

            Now substitute back in:

          So, the result is:

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. Let .

            Then let and substitute :

            1. The integral of is when :

            Now substitute back in:

          So, the result is:

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. The integral of cosine is sine:

          So, the result is:

        The result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Let .

        Then let and substitute :

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. The integral of is when :

          So, the result is:

        Now substitute back in:

      1. The integral of a constant times a function is the constant times the integral of the function:

        1. Let .

          Then let and substitute :

          1. The integral of a constant times a function is the constant times the integral of the function:

            1. The integral of is when :

            So, the result is:

          Now substitute back in:

        So, the result is:

      1. Let .

        Then let and substitute :

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. The integral of cosine is sine:

          So, the result is:

        Now substitute back in:

      The result is:

    Method #3

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Let .

        Then let and substitute :

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. The integral of is when :

          So, the result is:

        Now substitute back in:

      1. The integral of a constant times a function is the constant times the integral of the function:

        1. Let .

          Then let and substitute :

          1. The integral of a constant times a function is the constant times the integral of the function:

            1. The integral of is when :

            So, the result is:

          Now substitute back in:

        So, the result is:

      1. Let .

        Then let and substitute :

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. The integral of cosine is sine:

          So, the result is:

        Now substitute back in:

      The result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                   
 |                                  3           5     
 |    5               sin(2*x)   sin (2*x)   sin (2*x)
 | cos (2*x) dx = C + -------- - --------- + ---------
 |                       2           3           10   
/                                                     
$${{{{\sin ^5\left(2\,x\right)}\over{5}}-{{2\,\sin ^3\left(2\,x \right)}\over{3}}+\sin \left(2\,x\right)}\over{2}}$$
The graph
The answer [src]
            3         5   
sin(2)   sin (2)   sin (2)
------ - ------- + -------
  2         3         10  
$${{3\,\sin ^52-10\,\sin ^32+15\,\sin 2}\over{30}}$$
=
=
            3         5   
sin(2)   sin (2)   sin (2)
------ - ------- + -------
  2         3         10  
$$- \frac{\sin^{3}{\left(2 \right)}}{3} + \frac{\sin^{5}{\left(2 \right)}}{10} + \frac{\sin{\left(2 \right)}}{2}$$
Numerical answer [src]
0.266202423408773
0.266202423408773
The graph
Integral of cos^5(2x) dx

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