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x^2sin(x)^2

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

Limits of integration:

from to
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The graph:

from to

Piecewise:

The solution

You have entered [src]
  1              
  /              
 |               
 |   2    2      
 |  x *sin (x) dx
 |               
/                
0                
$$\int\limits_{0}^{1} x^{2} \sin^{2}{\left(x \right)}\, dx$$
Integral(x^2*sin(x)^2, (x, 0, 1))
Detail solution
  1. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. The integral of a constant is the constant times the variable of integration:

      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 a constant times a function is the constant times the integral of the function:

            1. The integral of cosine is sine:

            So, the result is:

          Now substitute back in:

        So, the result is:

      The result is:

    Now evaluate the sub-integral.

  2. Use integration by parts:

    Let and let .

    Then .

    To find :

    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 a constant times a function is the constant times the integral of the function:

        1. There are multiple ways to do this integral.

          Method #1

          1. Let .

            Then let and substitute :

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

              1. The integral of sine is negative cosine:

              So, the result is:

            Now substitute back in:

          Method #2

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

            1. There are multiple ways to do this integral.

              Method #1

              1. Let .

                Then let and substitute :

                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:

                Now substitute back in:

              Method #2

              1. Let .

                Then let and substitute :

                1. The integral of is when :

                Now substitute back in:

            So, the result is:

        So, the result is:

      The result is:

    Now evaluate the sub-integral.

  3. 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 a constant times a function is the constant times the integral of the function:

      1. Let .

        Then let and substitute :

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

          1. The integral of cosine is sine:

          So, the result is:

        Now substitute back in:

      So, the result is:

    The result is:

  4. Now simplify:

  5. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                          / 2   cos(2*x)\
 |                      3                                  x*|x  + --------|
 |  2    2             x    sin(2*x)    2 /x   sin(2*x)\     \        2    /
 | x *sin (x) dx = C + -- + -------- + x *|- - --------| - -----------------
 |                     6       8          \2      4    /           2        
/                                                                           
$$-{{\left(6\,x^2-3\right)\,\sin \left(2\,x\right)+6\,x\,\cos \left(2 \,x\right)-4\,x^3}\over{24}}$$
The graph
The answer [src]
     2           2                   
  cos (1)   5*sin (1)   cos(1)*sin(1)
- ------- + --------- - -------------
     12         12            4      
$$-{{3\,\sin 2+6\,\cos 2-4}\over{24}}$$
=
=
     2           2                   
  cos (1)   5*sin (1)   cos(1)*sin(1)
- ------- + --------- - -------------
     12         12            4      
$$- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{4} - \frac{\cos^{2}{\left(1 \right)}}{12} + \frac{5 \sin^{2}{\left(1 \right)}}{12}$$
Numerical answer [src]
0.157041197450242
0.157041197450242
The graph
Integral of x^2sin(x)^2 dx

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