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Integral of cos^5xsinx
Identical expressions
cos^5xsinx
co sinus of e of to the power of 5x sinus of x
cos5xsinx
cos⁵xsinx
cos^5xsinxdx

Solving integrals/ cos^5xsinx

Integral of cos^5xsinx dx

The solution

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 pi                  
 --                  
 2                   
  /                  
 |                   
 |     5             
 |  cos (x)*sin(x) dx
 |                   
/                    
1

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

Integral(cos(x)^5*sin(x), (x, 1, pi/2))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \cos{\left(x \right)}$ .
  
  Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
  
  $\int \left(- u^{5}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int u^{5}\, du = - \int u^{5}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{5}\, du = \frac{u^{6}}{6}$
    So, the result is: $- \frac{u^{6}}{6}$
  Now substitute $u$ back in:
  
  $- \frac{\cos^{6}{\left(x \right)}}{6}$
Method #2
1. Rewrite the integrand:
  
  $\sin{\left(x \right)} \cos^{5}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos{\left(x \right)}$
2. Let $u = 1 - \sin^{2}{\left(x \right)}$ .
  
  Then let $du = - 2 \sin{\left(x \right)} \cos{\left(x \right)} dx$ and substitute $- \frac{du}{2}$ :
  
  $\int \left(- \frac{u^{2}}{2}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int u^{2}\, du = - \frac{\int u^{2}\, du}{2}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
    So, the result is: $- \frac{u^{3}}{6}$
  Now substitute $u$ back in:
  
  $- \frac{\left(1 - \sin^{2}{\left(x \right)}\right)^{3}}{6}$
Add the constant of integration:

$- \frac{\cos^{6}{\left(x \right)}}{6}+ \mathrm{constant}$

The answer is:

$- \frac{\cos^{6}{\left(x \right)}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                            6   
 |    5                    cos (x)
 | cos (x)*sin(x) dx = C - -------
 |                            6   
/

$$\int \sin{\left(x \right)} \cos^{5}{\left(x \right)}\, dx = C - \frac{\cos^{6}{\left(x \right)}}{6}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   6   
cos (1)
-------
   6

$$\frac{\cos^{6}{\left(1 \right)}}{6}$$

   6   
cos (1)
-------
   6

$$\frac{\cos^{6}{\left(1 \right)}}{6}$$

cos(1)^6/6

Numerical answer [src]

0.00414638548573729

0.00414638548573729

The graph

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

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Identical expressions

Integral of cos^5xsinx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2