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sin3sin2xdx
sin^3sin2xdx

Integral of sin^3*sin(2x)dx dx

The solution

You have entered [src]

  1                    
  /                    
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 |     3               
 |  sin (x)*sin(2*x) dx
 |                     
/                      
0

$$\int\limits_{0}^{1} \sin^{3}{\left(x \right)} \sin{\left(2 x \right)}\, dx$$

Integral(sin(x)^3*sin(2*x), (x, 0, 1))

Detail solution

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

$\frac{2 \sin^{5}{\left(x \right)}}{5}+ \mathrm{constant}$

The answer is:

$\frac{2 \sin^{5}{\left(x \right)}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                5   
 |    3                      2*sin (x)
 | sin (x)*sin(2*x) dx = C + ---------
 |                               5    
/

$$\int \sin^{3}{\left(x \right)} \sin{\left(2 x \right)}\, dx = C + \frac{2 \sin^{5}{\left(x \right)}}{5}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       3                  3                  2                       2                 
  2*sin (1)*cos(2)   2*cos (1)*sin(2)   4*cos (1)*cos(2)*sin(1)   sin (1)*cos(1)*sin(2)
- ---------------- + ---------------- - ----------------------- - ---------------------
         5                  5                      5                        5

$$- \frac{\sin^{2}{\left(1 \right)} \sin{\left(2 \right)} \cos{\left(1 \right)}}{5} + \frac{2 \sin{\left(2 \right)} \cos^{3}{\left(1 \right)}}{5} - \frac{4 \sin{\left(1 \right)} \cos^{2}{\left(1 \right)} \cos{\left(2 \right)}}{5} - \frac{2 \sin^{3}{\left(1 \right)} \cos{\left(2 \right)}}{5}$$

       3                  3                  2                       2                 
  2*sin (1)*cos(2)   2*cos (1)*sin(2)   4*cos (1)*cos(2)*sin(1)   sin (1)*cos(1)*sin(2)
- ---------------- + ---------------- - ----------------------- - ---------------------
         5                  5                      5                        5

-2*sin(1)^3*cos(2)/5 + 2*cos(1)^3*sin(2)/5 - 4*cos(1)^2*cos(2)*sin(1)/5 - sin(1)^2*cos(1)*sin(2)/5

Numerical answer [src]

0.168754638327912

0.168754638327912

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

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How to use it?

Identical expressions

Integral of sin^3*sin(2x)dx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2