Mister Exam

Integral of xcos(pix) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                            
 |                      cos(pi*x)   x*sin(pi*x)
 | x*cos(pi*x) dx = C + --------- + -----------
 |                           2           pi    
/                          pi                  
$$\int x \cos{\left(\pi x \right)}\, dx = C + \frac{x \sin{\left(\pi x \right)}}{\pi} + \frac{\cos{\left(\pi x \right)}}{\pi^{2}}$$
The graph
The answer [src]
-2 
---
  2
pi 
$$- \frac{2}{\pi^{2}}$$
=
=
-2 
---
  2
pi 
$$- \frac{2}{\pi^{2}}$$
-2/pi^2
Numerical answer [src]
-0.202642367284676
-0.202642367284676
The graph
Integral of xcos(pix) dx

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