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Integral of sin3xsinxsin(x) dx

The solution

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 oo                          
  /                          
 |                           
 |  sin(3*x)*sin(x)*sin(x) dx
 |                           
/                            
0

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

Integral((sin(3*x)*sin(x))*sin(x), (x, 0, oo))

Detail solution

Rewrite the integrand:

$\sin{\left(x \right)} \sin{\left(3 x \right)} \sin{\left(x \right)} = - 4 \sin^{5}{\left(x \right)} + 3 \sin^{3}{\left(x \right)}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 4 \sin^{5}{\left(x \right)}\right)\, dx = - 4 \int \sin^{5}{\left(x \right)}\, dx$
  1. Rewrite the integrand:
    
    $\sin^{5}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)}$
  2. There are multiple ways to do this integral.
    Method #1
    1. Rewrite the integrand:
      
      $\left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} = \sin{\left(x \right)} \cos^{4}{\left(x \right)} - 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \sin{\left(x \right)}$
    2. Integrate term-by-term:
      
      Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int \left(- u^{4}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{4}\, du = - \int u^{4}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
      
      So, the result is: $- \frac{u^{5}}{5}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{5}{\left(x \right)}}{5}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 2 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$
      
      Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int \left(- u^{2}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{2}\, du = - \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $- \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{3}{\left(x \right)}}{3}$
      
      So, the result is: $\frac{2 \cos^{3}{\left(x \right)}}{3}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
      
      The result is: $- \frac{\cos^{5}{\left(x \right)}}{5} + \frac{2 \cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$
    Method #2
    1. Rewrite the integrand:
      
      $\left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} = \sin{\left(x \right)} \cos^{4}{\left(x \right)} - 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \sin{\left(x \right)}$
    2. Integrate term-by-term:
      
      Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int \left(- u^{4}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{4}\, du = - \int u^{4}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
      
      So, the result is: $- \frac{u^{5}}{5}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{5}{\left(x \right)}}{5}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 2 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$
      
      Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int \left(- u^{2}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{2}\, du = - \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $- \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{3}{\left(x \right)}}{3}$
      
      So, the result is: $\frac{2 \cos^{3}{\left(x \right)}}{3}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
      
      The result is: $- \frac{\cos^{5}{\left(x \right)}}{5} + \frac{2 \cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$
  So, the result is: $\frac{4 \cos^{5}{\left(x \right)}}{5} - \frac{8 \cos^{3}{\left(x \right)}}{3} + 4 \cos{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 3 \sin^{3}{\left(x \right)}\, dx = 3 \int \sin^{3}{\left(x \right)}\, dx$
  1. Rewrite the integrand:
    
    $\sin^{3}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}$
  2. Let $u = \cos{\left(x \right)}$ .
    
    Then let $du = - \sin{\left(x \right)} dx$ and substitute $du$ :
    
    $\int \left(u^{2} - 1\right)\, du$
    1. Integrate term-by-term:
      1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int u^{2}\, du = \frac{u^{3}}{3}$
      1. The integral of a constant is the constant times the variable of integration:
        
        $\int \left(-1\right)\, du = - u$
      The result is: $\frac{u^{3}}{3} - u$
    Now substitute $u$ back in:
    
    $\frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$
  So, the result is: $\cos^{3}{\left(x \right)} - 3 \cos{\left(x \right)}$
The result is: $\frac{4 \cos^{5}{\left(x \right)}}{5} - \frac{5 \cos^{3}{\left(x \right)}}{3} + \cos{\left(x \right)}$
Now simplify:

$\frac{\left(12 \sin^{4}{\left(x \right)} + \sin^{2}{\left(x \right)} + 2\right) \cos{\left(x \right)}}{15}$
Add the constant of integration:

$\frac{\left(12 \sin^{4}{\left(x \right)} + \sin^{2}{\left(x \right)} + 2\right) \cos{\left(x \right)}}{15}+ \mathrm{constant}$

The answer is:

$\frac{\left(12 \sin^{4}{\left(x \right)} + \sin^{2}{\left(x \right)} + 2\right) \cos{\left(x \right)}}{15}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

 -17   13 
<----, -->
  15   15

$$\left\langle - \frac{17}{15}, \frac{13}{15}\right\rangle$$

 -17   13 
<----, -->
  15   15

$$\left\langle - \frac{17}{15}, \frac{13}{15}\right\rangle$$

AccumBounds(-17/15, 13/15)

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of sin3xsinxsin(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

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Integral of sin3x*sinx*sin(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Integral of sin3xsinxsin(x) dx