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Solving integrals/ exp(y-x)*sin(y)

Integral of exp(y-x)*sin(y) dx

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The solution

You have entered [src] 
  y                 
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 |   y - x          
 |  e     *sin(y) dx
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0
∫0ye−x+ysin⁡(y) dx\int\limits_{0}^{y} e^{- x + y} \sin{\left(y \right)}\, dx 
Integral(exp(y - x)*sin(y), (x, 0, y))
Detail solution

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

    ∫e−x+ysin⁡(y) dx=sin⁡(y)∫e−x+y dx\int e^{- x + y} \sin{\left(y \right)}\, dx = \sin{\left(y \right)} \int e^{- x + y}\, dx

    1. Let u=−x+yu = - x + y.

      Then let du=−dxdu = - dx and substitute −du- du:

      ∫(−eu) du\int \left(- e^{u}\right)\, du

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

        False\text{False}

        1. The integral of the exponential function is itself.

          ∫eu du=eu\int e^{u}\, du = e^{u}

        So, the result is: −eu- e^{u}

      Now substitute uu back in:

      −e−x+y- e^{- x + y}

    So, the result is: −e−x+ysin⁡(y)- e^{- x + y} \sin{\left(y \right)}

  2. Add the constant of integration:

    −e−x+ysin⁡(y)+constant- e^{- x + y} \sin{\left(y \right)}+ \mathrm{constant}

The answer is:

−e−x+ysin⁡(y)+constant- e^{- x + y} \sin{\left(y \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                    
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 |  y - x                  y - x       
 | e     *sin(y) dx = C - e     *sin(y)
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∫e−x+ysin⁡(y) dx=C−e−x+ysin⁡(y)\int e^{- x + y} \sin{\left(y \right)}\, dx = C - e^{- x + y} \sin{\left(y \right)}
Simplify
The answer [src] 
           y       
-sin(y) + e *sin(y)
eysin⁡(y)−sin⁡(y)e^{y} \sin{\left(y \right)} - \sin{\left(y \right)} 
=
= 
           y       
-sin(y) + e *sin(y)
eysin⁡(y)−sin⁡(y)e^{y} \sin{\left(y \right)} - \sin{\left(y \right)} 
-sin(y) + exp(y)*sin(y)

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

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