Integral of (cos⁵x)½ dx

The solution

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  1           
  /           
 |            
 |     5      
 |  cos (x)   
 |  ------- dx
 |     2      
 |            
/             
0

$$\int\limits_{0}^{1} \frac{\cos^{5}{\left(x \right)}}{2}\, dx$$

Integral(cos(x)^5/2, (x, 0, 1))

Detail solution

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

$\int \frac{\cos^{5}{\left(x \right)}}{2}\, dx = \frac{\int \cos^{5}{\left(x \right)}\, dx}{2}$
1. Rewrite the integrand:
  
  $\cos^{5}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right)^{2} \cos{\left(x \right)}$
2. There are multiple ways to do this integral.
  Method #1
  1. Rewrite the integrand:
    
    $\left(1 - \sin^{2}{\left(x \right)}\right)^{2} \cos{\left(x \right)} = \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 2 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$
  2. Integrate term-by-term:
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{4}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{5}{\left(x \right)}}{5}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 2 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
      
      Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{3}{\left(x \right)}}{3}$
      
      So, the result is: $- \frac{2 \sin^{3}{\left(x \right)}}{3}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
    The result is: $\frac{\sin^{5}{\left(x \right)}}{5} - \frac{2 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$
  Method #2
  1. Rewrite the integrand:
    
    $\left(1 - \sin^{2}{\left(x \right)}\right)^{2} \cos{\left(x \right)} = \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 2 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$
  2. Integrate term-by-term:
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{4}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{5}{\left(x \right)}}{5}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 2 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
      
      Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{3}{\left(x \right)}}{3}$
      
      So, the result is: $- \frac{2 \sin^{3}{\left(x \right)}}{3}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
    The result is: $\frac{\sin^{5}{\left(x \right)}}{5} - \frac{2 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$
So, the result is: $\frac{\sin^{5}{\left(x \right)}}{10} - \frac{\sin^{3}{\left(x \right)}}{3} + \frac{\sin{\left(x \right)}}{2}$
Add the constant of integration:

$\frac{\sin^{5}{\left(x \right)}}{10} - \frac{\sin^{3}{\left(x \right)}}{3} + \frac{\sin{\left(x \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\sin^{5}{\left(x \right)}}{10} - \frac{\sin^{3}{\left(x \right)}}{3} + \frac{\sin{\left(x \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                           
 |                                            
 |    5                         3         5   
 | cos (x)          sin(x)   sin (x)   sin (x)
 | ------- dx = C + ------ - ------- + -------
 |    2               2         3         10  
 |                                            
/

$$\int \frac{\cos^{5}{\left(x \right)}}{2}\, dx = C + \frac{\sin^{5}{\left(x \right)}}{10} - \frac{\sin^{3}{\left(x \right)}}{3} + \frac{\sin{\left(x \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

            3         5   
sin(1)   sin (1)   sin (1)
------ - ------- + -------
  2         3         10

$$- \frac{\sin^{3}{\left(1 \right)}}{3} + \frac{\sin^{5}{\left(1 \right)}}{10} + \frac{\sin{\left(1 \right)}}{2}$$

            3         5   
sin(1)   sin (1)   sin (1)
------ - ------- + -------
  2         3         10

$$- \frac{\sin^{3}{\left(1 \right)}}{3} + \frac{\sin^{5}{\left(1 \right)}}{10} + \frac{\sin{\left(1 \right)}}{2}$$

sin(1)/2 - sin(1)^3/3 + sin(1)^5/10

Numerical answer [src]

0.264316406455608

0.264316406455608

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

Other calculators

How to use it?

Identical expressions

Integral of (cos⁵x)½ dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2