Mister Exam

Integral of xsin(x+y) dx

Limits of integration:

from to
v

The graph:

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Piecewise:

The solution

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01xsin(x+y)dx\int\limits_{0}^{1} x \sin{\left(x + y \right)}\, dx
Integral(x*sin(x + y), (x, 0, 1))
Detail solution
  1. Use integration by parts:

    udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=xu{\left(x \right)} = x and let dv(x)=sin(x+y)\operatorname{dv}{\left(x \right)} = \sin{\left(x + y \right)}.

    Then du(x)=1\operatorname{du}{\left(x \right)} = 1.

    To find v(x)v{\left(x \right)}:

    1. Let u=x+yu = x + y.

      Then let du=dxdu = dx and substitute dudu:

      sin(u)du\int \sin{\left(u \right)}\, du

      1. The integral of sine is negative cosine:

        sin(u)du=cos(u)\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}

      Now substitute uu back in:

      cos(x+y)- \cos{\left(x + y \right)}

    Now evaluate the sub-integral.

  2. The integral of a constant times a function is the constant times the integral of the function:

    (cos(x+y))dx=cos(x+y)dx\int \left(- \cos{\left(x + y \right)}\right)\, dx = - \int \cos{\left(x + y \right)}\, dx

    1. Let u=x+yu = x + y.

      Then let du=dxdu = dx and substitute dudu:

      cos(u)du\int \cos{\left(u \right)}\, du

      1. The integral of cosine is sine:

        cos(u)du=sin(u)\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}

      Now substitute uu back in:

      sin(x+y)\sin{\left(x + y \right)}

    So, the result is: sin(x+y)- \sin{\left(x + y \right)}

  3. Add the constant of integration:

    xcos(x+y)+sin(x+y)+constant- x \cos{\left(x + y \right)} + \sin{\left(x + y \right)}+ \mathrm{constant}


The answer is:

xcos(x+y)+sin(x+y)+constant- x \cos{\left(x + y \right)} + \sin{\left(x + y \right)}+ \mathrm{constant}

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

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