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xcos(x)^2

Integral of xcos(x)^2 dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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

  2. 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:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                   
 |                     2                              
 |      2             x    cos(2*x)     /x   sin(2*x)\
 | x*cos (x) dx = C - -- + -------- + x*|- + --------|
 |                    4       8         \2      4    /
/                                                     
$${{2\,x\,\sin \left(2\,x\right)+\cos \left(2\,x\right)+2\,x^2}\over{ 8}}$$
The graph
The answer [src]
   2                   
cos (1)   cos(1)*sin(1)
------- + -------------
   4            2      
$${{2\,\sin 2+\cos 2+2}\over{8}}-{{1}\over{8}}$$
=
=
   2                   
cos (1)   cos(1)*sin(1)
------- + -------------
   4            2      
$$\frac{\cos^{2}{\left(1 \right)}}{4} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2}$$
Numerical answer [src]
0.300306002138028
0.300306002138028
The graph
Integral of xcos(x)^2 dx

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