Mister Exam

Integral of xcos(x-y) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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

      Now substitute back in:

    Now evaluate the sub-integral.

  2. Let .

    Then let and substitute :

    1. The integral of sine is negative cosine:

    Now substitute back in:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                               
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 | x*cos(x - y) dx = C + x*sin(x - y) + cos(x - y)
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$$\left(y-x\right)\,\sin \left(y-x\right)-y\,\sin \left(y-x\right)+ \cos \left(y-x\right)$$
The answer [src]
-cos(y) - sin(-1 + y) + cos(-1 + y)
$$-\cos y-\sin \left(y-1\right)+\cos \left(y-1\right)$$
=
=
-cos(y) - sin(-1 + y) + cos(-1 + y)
$$- \sin{\left(y - 1 \right)} - \cos{\left(y \right)} + \cos{\left(y - 1 \right)}$$

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