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Solving integrals/ (cos(x)-xsin(x))y

Integral of (cos(x)-xsin(x))y dx

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The solution

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x0
∫x0xy(−xsin⁡(x)+cos⁡(x)) dx\int\limits_{x_{0}}^{x} y \left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right)\, dx 
Integral((cos(x) - x*sin(x))*y, (x, x0, x))
Detail solution

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

    ∫y(−xsin⁡(x)+cos⁡(x)) dx=y∫(−xsin⁡(x)+cos⁡(x)) dx\int y \left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right)\, dx = y \int \left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right)\, dx

    1. Integrate term-by-term:

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

        ∫(−xsin⁡(x)) dx=−∫xsin⁡(x) dx\int \left(- x \sin{\left(x \right)}\right)\, dx = - \int x \sin{\left(x \right)}\, dx

        1. Use integration by parts:

          ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\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)\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}.

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

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

          1. The integral of sine is negative cosine:

            ∫sin⁡(x) dx=−cos⁡(x)\int \sin{\left(x \right)}\, dx = - \cos{\left(x \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)) dx=−∫cos⁡(x) dx\int \left(- \cos{\left(x \right)}\right)\, dx = - \int \cos{\left(x \right)}\, dx

          1. The integral of cosine is sine:

            ∫cos⁡(x) dx=sin⁡(x)\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}

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

        So, the result is: xcos⁡(x)−sin⁡(x)x \cos{\left(x \right)} - \sin{\left(x \right)}

      1. The integral of cosine is sine:

        ∫cos⁡(x) dx=sin⁡(x)\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}

      The result is: xcos⁡(x)x \cos{\left(x \right)}

    So, the result is: xycos⁡(x)x y \cos{\left(x \right)}

  2. Add the constant of integration:

    xycos⁡(x)+constantx y \cos{\left(x \right)}+ \mathrm{constant}

The answer is:

xycos⁡(x)+constantx y \cos{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
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 | (cos(x) - x*sin(x))*y dx = C + x*y*cos(x)
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∫y(−xsin⁡(x)+cos⁡(x)) dx=C+xycos⁡(x)\int y \left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right)\, dx = C + x y \cos{\left(x \right)}
Simplify
The answer [src] 
x*y*cos(x) - x0*y*cos(x0)
xycos⁡(x)−x0ycos⁡(x0)x y \cos{\left(x \right)} - x_{0} y \cos{\left(x_{0} \right)} 
=
= 
x*y*cos(x) - x0*y*cos(x0)
xycos⁡(x)−x0ycos⁡(x0)x y \cos{\left(x \right)} - x_{0} y \cos{\left(x_{0} \right)} 
x*y*cos(x) - x0*y*cos(x0)

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

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