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x^2cos2x

Integral of x^2cos2x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1               
  /               
 |                
 |   2            
 |  x *cos(2*x) dx
 |                
/                 
0                 
$$\int\limits_{0}^{1} x^{2} \cos{\left(2 x \right)}\, dx$$
Integral(x^2*cos(2*x), (x, 0, 1))
Detail solution
  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 cosine is sine:

        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. 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:

    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 cosine is sine:

        So, the result is:

      Now substitute back in:

    So, the result is:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                        
 |                                               2         
 |  2                   sin(2*x)   x*cos(2*x)   x *sin(2*x)
 | x *cos(2*x) dx = C - -------- + ---------- + -----------
 |                         4           2             2     
/                                                          
$$\int x^{2} \cos{\left(2 x \right)}\, dx = C + \frac{x^{2} \sin{\left(2 x \right)}}{2} + \frac{x \cos{\left(2 x \right)}}{2} - \frac{\sin{\left(2 x \right)}}{4}$$
The graph
The answer [src]
cos(2)   sin(2)
------ + ------
  2        4   
$$\frac{\cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)}}{4}$$
=
=
cos(2)   sin(2)
------ + ------
  2        4   
$$\frac{\cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)}}{4}$$
cos(2)/2 + sin(2)/4
Numerical answer [src]
0.0192509384328492
0.0192509384328492
The graph
Integral of x^2cos2x dx

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