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Integral of cos(x)*sin^3x dx

The solution

You have entered [src]

 pi                  
 --                  
 3                   
  /                  
 |                   
 |            3      
 |  cos(x)*sin (x) dx
 |                   
/                    
0

$$\int\limits_{0}^{\frac{\pi}{3}} \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx$$

Integral(cos(x)*sin(x)^3, (x, 0, pi/3))

Detail solution

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

$\frac{\sin^{4}{\left(x \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{\sin^{4}{\left(x \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                            4   
 |           3             sin (x)
 | cos(x)*sin (x) dx = C + -------
 |                            4   
/

$$\int \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx = C + \frac{\sin^{4}{\left(x \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

9/64

$$\frac{9}{64}$$

9/64

$$\frac{9}{64}$$

9/64

Numerical answer [src]

0.140625

0.140625

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

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Identical expressions

Similar expressions

Integral of cos(x)*sin^3x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2