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Solving integrals/ sin^3xsin2x

Integral of sin^3xsin2x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  n                    
  -                    
  2                    
  /                    
 |                     
 |     3               
 |  sin (x)*sin(2*x) dx
 |                     
/                      
0
$$\int\limits_{0}^{\frac{n}{2}} \sin^{3}{\left(x \right)} \sin{\left(2 x \right)}\, dx$$ 
Integral(sin(x)^3*sin(2*x), (x, 0, n/2))
Detail solution

  1. 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:

      1. Let .

        Then let and substitute :

        1. The integral of is when :

        Now substitute back in:

      So, the result is:

    Method #2

    1. Rewrite the integrand:

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

      1. Let .

        Then let and substitute :

        1. The integral of is when :

        Now substitute back in:

      So, the result is:

  2. Add the constant of integration:

The answer is:

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 answer [src] 
       3/n\               3/n\               2/n\           /n\      2/n\    /n\       
  2*sin |-|*cos(n)   2*cos |-|*sin(n)   4*cos |-|*cos(n)*sin|-|   sin |-|*cos|-|*sin(n)
        \2/                \2/                \2/           \2/       \2/    \2/       
- ---------------- + ---------------- - ----------------------- - ---------------------
         5                  5                      5                        5
$$- \frac{2 \sin^{3}{\left(\frac{n}{2} \right)} \cos{\left(n \right)}}{5} - \frac{\sin^{2}{\left(\frac{n}{2} \right)} \sin{\left(n \right)} \cos{\left(\frac{n}{2} \right)}}{5} - \frac{4 \sin{\left(\frac{n}{2} \right)} \cos^{2}{\left(\frac{n}{2} \right)} \cos{\left(n \right)}}{5} + \frac{2 \sin{\left(n \right)} \cos^{3}{\left(\frac{n}{2} \right)}}{5}$$ 
=
= 
       3/n\               3/n\               2/n\           /n\      2/n\    /n\       
  2*sin |-|*cos(n)   2*cos |-|*sin(n)   4*cos |-|*cos(n)*sin|-|   sin |-|*cos|-|*sin(n)
        \2/                \2/                \2/           \2/       \2/    \2/       
- ---------------- + ---------------- - ----------------------- - ---------------------
         5                  5                      5                        5
$$- \frac{2 \sin^{3}{\left(\frac{n}{2} \right)} \cos{\left(n \right)}}{5} - \frac{\sin^{2}{\left(\frac{n}{2} \right)} \sin{\left(n \right)} \cos{\left(\frac{n}{2} \right)}}{5} - \frac{4 \sin{\left(\frac{n}{2} \right)} \cos^{2}{\left(\frac{n}{2} \right)} \cos{\left(n \right)}}{5} + \frac{2 \sin{\left(n \right)} \cos^{3}{\left(\frac{n}{2} \right)}}{5}$$ 
-2*sin(n/2)^3*cos(n)/5 + 2*cos(n/2)^3*sin(n)/5 - 4*cos(n/2)^2*cos(n)*sin(n/2)/5 - sin(n/2)^2*cos(n/2)*sin(n)/5

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

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