Mister Exam

Integral of cos⁵xdx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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

  2. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Let .

        Then let and substitute :

        1. The integral of is when :

        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 is when :

          Now substitute back in:

        So, the result is:

      1. The integral of cosine is sine:

      The result is:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Let .

        Then let and substitute :

        1. The integral of is when :

        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 is when :

          Now substitute back in:

        So, the result is:

      1. The integral of cosine is sine:

      The result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                               
 |                         3         5            
 |    5               2*sin (x)   sin (x)         
 | cos (x)*1 dx = C - --------- + ------- + sin(x)
 |                        3          5            
/                                                 
$${{\sin ^5x}\over{5}}-{{2\,\sin ^3x}\over{3}}+\sin x$$
The graph
The answer [src]
       3         5            
  2*sin (1)   sin (1)         
- --------- + ------- + sin(1)
      3          5            
$${{3\,\sin ^51-10\,\sin ^31+15\,\sin 1}\over{15}}$$
=
=
       3         5            
  2*sin (1)   sin (1)         
- --------- + ------- + sin(1)
      3          5            
$$- \frac{2 \sin^{3}{\left(1 \right)}}{3} + \frac{\sin^{5}{\left(1 \right)}}{5} + \sin{\left(1 \right)}$$
Numerical answer [src]
0.528632812911216
0.528632812911216
The graph
Integral of cos⁵xdx dx

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