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Identical expressions
(sin(3x))^ four *cos(3x)
( sinus of (3x)) to the power of 4 multiply by co sinus of e of (3x)
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(sin(3x))⁴*cos(3x)
(sin(3x))^4cos(3x)
(sin(3x))4cos(3x)
sin3x4cos3x
sin3x^4cos3x
(sin(3x))^4*cos(3x)dx

Integral of (sin(3x))^4*cos(3x) dx

The solution

You have entered [src]

  1                      
  /                      
 |                       
 |     4                 
 |  sin (3*x)*cos(3*x) dx
 |                       
/                        
0

$$\int\limits_{0}^{1} \sin^{4}{\left(3 x \right)} \cos{\left(3 x \right)}\, dx$$

Integral(sin(3*x)^4*cos(3*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sin{\left(3 x \right)}$ .
  
  Then let $du = 3 \cos{\left(3 x \right)} dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{u^{4}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int u^{4}\, du = \frac{\int u^{4}\, du}{3}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
    So, the result is: $\frac{u^{5}}{15}$
  Now substitute $u$ back in:
  
  $\frac{\sin^{5}{\left(3 x \right)}}{15}$
Method #2
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{\sin^{4}{\left(u \right)} \cos{\left(u \right)}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin^{4}{\left(u \right)} \cos{\left(u \right)}\, du = \frac{\int \sin^{4}{\left(u \right)} \cos{\left(u \right)}\, du}{3}$
    1. Let $u = \sin{\left(u \right)}$ .
      
      Then let $du = \cos{\left(u \right)} du$ 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:
      
      $\frac{\sin^{5}{\left(u \right)}}{5}$
    So, the result is: $\frac{\sin^{5}{\left(u \right)}}{15}$
  Now substitute $u$ back in:
  
  $\frac{\sin^{5}{\left(3 x \right)}}{15}$
Add the constant of integration:

$\frac{\sin^{5}{\left(3 x \right)}}{15}+ \mathrm{constant}$

The answer is:

$\frac{\sin^{5}{\left(3 x \right)}}{15}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     
 |                                5     
 |    4                        sin (3*x)
 | sin (3*x)*cos(3*x) dx = C + ---------
 |                                 15   
/

$$\int \sin^{4}{\left(3 x \right)} \cos{\left(3 x \right)}\, dx = C + \frac{\sin^{5}{\left(3 x \right)}}{15}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   5   
sin (3)
-------
   15

$$\frac{\sin^{5}{\left(3 \right)}}{15}$$

   5   
sin (3)
-------
   15

$$\frac{\sin^{5}{\left(3 \right)}}{15}$$

sin(3)^5/15

Numerical answer [src]

3.73122727919266e-6

3.73122727919266e-6

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

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Identical expressions

Integral of (sin(3x))^4*cos(3x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2