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Integral of xcos((x-pi)/3) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  1                 
  /                 
 |                  
 |       /x - pi\   
 |  x*cos|------| dx
 |       \  3   /   
 |                  
/                   
0                   
$$\int\limits_{0}^{1} x \cos{\left(\frac{x - \pi}{3} \right)}\, dx$$
Integral(x*cos((x - pi)/3), (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 sine is negative cosine:

        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 cosine is sine:

        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 - pi\               /x   pi\          /x   pi\
 | x*cos|------| dx = C + 9*sin|- + --| - 3*x*cos|- + --|
 |      \  3   /               \3   6 /          \3   6 /
 |                                                       
/                                                        
$$\int x \cos{\left(\frac{x - \pi}{3} \right)}\, dx = C - 3 x \cos{\left(\frac{x}{3} + \frac{\pi}{6} \right)} + 9 \sin{\left(\frac{x}{3} + \frac{\pi}{6} \right)}$$
The graph
The answer [src]
  9        /1   pi\        /1   pi\
- - - 3*cos|- + --| + 9*sin|- + --|
  2        \3   6 /        \3   6 /
$$- \frac{9}{2} - 3 \cos{\left(\frac{1}{3} + \frac{\pi}{6} \right)} + 9 \sin{\left(\frac{1}{3} + \frac{\pi}{6} \right)}$$
=
=
  9        /1   pi\        /1   pi\
- - - 3*cos|- + --| + 9*sin|- + --|
  2        \3   6 /        \3   6 /
$$- \frac{9}{2} - 3 \cos{\left(\frac{1}{3} + \frac{\pi}{6} \right)} + 9 \sin{\left(\frac{1}{3} + \frac{\pi}{6} \right)}$$
-9/2 - 3*cos(1/3 + pi/6) + 9*sin(1/3 + pi/6)
Numerical answer [src]
0.338258415318288
0.338258415318288

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