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x*cos(x/2)

Integral of x*cos(x/2) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                       
 |                                        
 |      /x\               /x\          /x\
 | x*cos|-| dx = C + 4*cos|-| + 2*x*sin|-|
 |      \2/               \2/          \2/
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/                                         
$$\int x \cos{\left(\frac{x}{2} \right)}\, dx = C + 2 x \sin{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{2} \right)}$$
The graph
The answer [src]
-4 + 2*sin(1/2) + 4*cos(1/2)
$$-4 + 2 \sin{\left(\frac{1}{2} \right)} + 4 \cos{\left(\frac{1}{2} \right)}$$
=
=
-4 + 2*sin(1/2) + 4*cos(1/2)
$$-4 + 2 \sin{\left(\frac{1}{2} \right)} + 4 \cos{\left(\frac{1}{2} \right)}$$
-4 + 2*sin(1/2) + 4*cos(1/2)
Numerical answer [src]
0.469181324769897
0.469181324769897
The graph
Integral of x*cos(x/2) dx

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