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cos³xsin²x

Integral cos³xsin²x dx

Limits of integration:

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The graph:

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The solution

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  1                   
  /                   
 |                    
 |     3       2      
 |  cos (x)*sin (x) dx
 |                    
/                     
0                     
$$\int\limits_{0}^{1} \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}\, dx$$
Integral(cos(x)^3*sin(x)^2, (x, 0, 1))
Detail solution
  1. Rewrite the integrand:

  2. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      1. Integrate term-by-term:

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

          1. The integral of is when :

          So, the result is:

        1. The integral of is when :

        The result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      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:

      1. Let .

        Then let and substitute :

        1. The integral of is when :

        Now substitute back in:

      The result is:

    Method #3

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      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:

      1. Let .

        Then let and substitute :

        1. The integral of is when :

        Now substitute back in:

      The result is:

  3. Add the constant of integration:


The answer is:

The graph
The answer [src]
     5         3   
  sin (1)   sin (1)
- ------- + -------
     5         3   
$$- \frac{\sin^{5}{\left(1 \right)}}{5} + \frac{\sin^{3}{\left(1 \right)}}{3}$$
=
=
     5         3   
  sin (1)   sin (1)
- ------- + -------
     5         3   
$$- \frac{\sin^{5}{\left(1 \right)}}{5} + \frac{\sin^{3}{\left(1 \right)}}{3}$$
-sin(1)^5/5 + sin(1)^3/3
Numerical answer [src]
0.114230426366362
0.114230426366362
The answer (Indefinite) [src]
  /                                          
 |                             5         3   
 |    3       2             sin (x)   sin (x)
 | cos (x)*sin (x) dx = C - ------- + -------
 |                             5         3   
/                                            
$$\int \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}\, dx = C - \frac{\sin^{5}{\left(x \right)}}{5} + \frac{\sin^{3}{\left(x \right)}}{3}$$

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

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