Mister Exam

Integral of cos^5x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1           
  /           
 |            
 |     5      
 |  cos (x) dx
 |            
/             
0             
$$\int\limits_{0}^{1} \cos^{5}{\left(x \right)}\, dx$$
Integral(cos(x)^5, (x, 0, 1))
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) dx = C - --------- + ------- + sin(x)
 |                      3          5            
/                                               
$$\int \cos^{5}{\left(x \right)}\, dx = C + \frac{\sin^{5}{\left(x \right)}}{5} - \frac{2 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$$
The graph
The answer [src]
       3         5            
  2*sin (1)   sin (1)         
- --------- + ------- + sin(1)
      3          5            
$$- \frac{2 \sin^{3}{\left(1 \right)}}{3} + \frac{\sin^{5}{\left(1 \right)}}{5} + \sin{\left(1 \right)}$$
=
=
       3         5            
  2*sin (1)   sin (1)         
- --------- + ------- + sin(1)
      3          5            
$$- \frac{2 \sin^{3}{\left(1 \right)}}{3} + \frac{\sin^{5}{\left(1 \right)}}{5} + \sin{\left(1 \right)}$$
-2*sin(1)^3/3 + sin(1)^5/5 + sin(1)
Numerical answer [src]
0.528632812911216
0.528632812911216
The graph
Integral of cos^5x dx

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