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Integral of sin(x+y)+xcos(x+y) dx

Limits of integration:

from to
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The graph:

from to

Piecewise:

The solution

You have entered [src]
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 |  (sin(x + y) + x*cos(x + y)) dx
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$$\int\limits_{0}^{1} \left(x \cos{\left(x + y \right)} + \sin{\left(x + y \right)}\right)\, dx$$
Integral(sin(x + y) + x*cos(x + y), (x, 0, 1))
Detail solution
  1. Integrate term-by-term:

    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:

    1. Let .

      Then let and substitute :

      1. The integral of sine is negative cosine:

      Now substitute back in:

    The result is:

  2. Add the constant of integration:


The answer is:

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

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