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Identical expressions
Sin^ five *x/ two *cos*x/ two
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Sin to the power of five multiply by x divide by two multiply by co sinus of e of multiply by x divide by two
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Sin^5*x/2*cos*x/2dx

Solving integrals/ Sin^5*x/2*cos*x/2

Integral of Sin^5x/2cos*x/2 dx

The solution

You have entered [src]

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

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

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

Detail solution

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

$\int \frac{\sin^{5}{\left(x \right)} \cos{\left(x \right)}}{2 \cdot 2}\, dx = \frac{\int \sin^{5}{\left(x \right)} \cos{\left(x \right)}\, dx}{4}$
1. There are multiple ways to do this integral.
  Method #1
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{5}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{5}\, du = \frac{u^{6}}{6}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{6}{\left(x \right)}}{6}$
  Method #2
  1. Rewrite the integrand:
    
    $\sin^{5}{\left(x \right)} \cos{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos{\left(x \right)}$
  2. Let $u = 1 - \cos^{2}{\left(x \right)}$ .
    
    Then let $du = 2 \sin{\left(x \right)} \cos{\left(x \right)} dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{u^{2}}{4}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u^{2}}{2}\, du = \frac{\int u^{2}\, du}{2}$
      
      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:
    
    $\frac{\left(1 - \cos^{2}{\left(x \right)}\right)^{3}}{6}$
So, the result is: $\frac{\sin^{6}{\left(x \right)}}{24}$
Add the constant of integration:

$\frac{\sin^{6}{\left(x \right)}}{24}+ \mathrm{constant}$

The answer is:

$\frac{\sin^{6}{\left(x \right)}}{24}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                                
 |    5                       6   
 | sin (x)*cos(x)          sin (x)
 | -------------- dx = C + -------
 |      2*2                   24  
 |                                
/

$${{\sin ^6x}\over{24}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   6   
sin (1)
-------
   24

$${{\sin ^61}\over{24}}$$

   6   
sin (1)
-------
   24

$$\frac{\sin^{6}{\left(1 \right)}}{24}$$

Simplify

Numerical answer [src]

0.0147918887192384

0.0147918887192384

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of Sin^5x/2cos*x/2 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

Integral of Sin^5*x/2*cos*x/2 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Integral of Sin^5x/2cos*x/2 dx