Mister Exam

Other calculators

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                    
  /                    
 |                     
 |     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
  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    
/                                     
$$\int \sin^{3}{\left(x \right)} \sin{\left(2 x \right)}\, dx = C + \frac{2 \sin^{5}{\left(x \right)}}{5}$$
The graph
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          
$$- \frac{\sin^{2}{\left(1 \right)} \sin{\left(2 \right)} \cos{\left(1 \right)}}{5} + \frac{2 \sin{\left(2 \right)} \cos^{3}{\left(1 \right)}}{5} - \frac{4 \sin{\left(1 \right)} \cos^{2}{\left(1 \right)} \cos{\left(2 \right)}}{5} - \frac{2 \sin^{3}{\left(1 \right)} \cos{\left(2 \right)}}{5}$$
-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.