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x^2sin2x

Integral of x^2sin2x dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  1               
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 |   2            
 |  x *sin(2*x) dx
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$$\int\limits_{0}^{1} x^{2} \sin{\left(2 x \right)}\, dx$$
Integral(x^2*sin(2*x), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      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. 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:

          So, the result is:

      Now evaluate the sub-integral.

    2. Use integration by parts:

      Let and let .

      Then .

      To find :

      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:

      Now evaluate the sub-integral.

    3. 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 sine is negative cosine:

          So, the result is:

        Now substitute back in:

      So, the result is:

    Method #2

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

      1. Use integration by parts:

        Let and let .

        Then .

        To find :

        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:

        Now evaluate the sub-integral.

      2. 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 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:

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

          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 sine is negative cosine:

              So, the result is:

            Now substitute back in:

          So, the result is:

        The result is:

      So, the result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                        
 |                                               2         
 |  2                   cos(2*x)   x*sin(2*x)   x *cos(2*x)
 | x *sin(2*x) dx = C + -------- + ---------- - -----------
 |                         4           2             2     
/                                                          
$$\int x^{2} \sin{\left(2 x \right)}\, dx = C - \frac{x^{2} \cos{\left(2 x \right)}}{2} + \frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4}$$
The graph
The answer [src]
  1   sin(2)   cos(2)
- - + ------ - ------
  4     2        4   
$$- \frac{1}{4} - \frac{\cos{\left(2 \right)}}{4} + \frac{\sin{\left(2 \right)}}{2}$$
=
=
  1   sin(2)   cos(2)
- - + ------ - ------
  4     2        4   
$$- \frac{1}{4} - \frac{\cos{\left(2 \right)}}{4} + \frac{\sin{\left(2 \right)}}{2}$$
Numerical answer [src]
0.308685422549626
0.308685422549626
The graph
Integral of x^2sin2x dx

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