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Solving integrals/ (e^(x+sin(y)))*cos(y)

Integral of (e^(x+sin(y)))*cos(y) dy

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The solution

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 |   x + sin(y)          
 |  E          *cos(y) dy
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∫01ex+sin⁡(y)cos⁡(y) dy\int\limits_{0}^{1} e^{x + \sin{\left(y \right)}} \cos{\left(y \right)}\, dy 
Integral(E^(x + sin(y))*cos(y), (y, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=x+sin⁡(y)u = x + \sin{\left(y \right)}.

      Then let du=cos⁡(y)dydu = \cos{\left(y \right)} dy and substitute dudu:

      ∫eu du\int e^{u}\, du

      1. The integral of the exponential function is itself.

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

      Now substitute uu back in:

      ex+sin⁡(y)e^{x + \sin{\left(y \right)}}

    Method #2

    1. Rewrite the integrand:

      ex+sin⁡(y)cos⁡(y)=exesin⁡(y)cos⁡(y)e^{x + \sin{\left(y \right)}} \cos{\left(y \right)} = e^{x} e^{\sin{\left(y \right)}} \cos{\left(y \right)}

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

      ∫exesin⁡(y)cos⁡(y) dy=ex∫esin⁡(y)cos⁡(y) dy\int e^{x} e^{\sin{\left(y \right)}} \cos{\left(y \right)}\, dy = e^{x} \int e^{\sin{\left(y \right)}} \cos{\left(y \right)}\, dy

      1. Let u=sin⁡(y)u = \sin{\left(y \right)}.

        Then let du=cos⁡(y)dydu = \cos{\left(y \right)} dy and substitute dudu:

        ∫eu du\int e^{u}\, du

        1. The integral of the exponential function is itself.

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

        Now substitute uu back in:

        esin⁡(y)e^{\sin{\left(y \right)}}

      So, the result is: exesin⁡(y)e^{x} e^{\sin{\left(y \right)}}

    Method #3

    1. Rewrite the integrand:

      ex+sin⁡(y)cos⁡(y)=exesin⁡(y)cos⁡(y)e^{x + \sin{\left(y \right)}} \cos{\left(y \right)} = e^{x} e^{\sin{\left(y \right)}} \cos{\left(y \right)}

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

      ∫exesin⁡(y)cos⁡(y) dy=ex∫esin⁡(y)cos⁡(y) dy\int e^{x} e^{\sin{\left(y \right)}} \cos{\left(y \right)}\, dy = e^{x} \int e^{\sin{\left(y \right)}} \cos{\left(y \right)}\, dy

      1. Let u=sin⁡(y)u = \sin{\left(y \right)}.

        Then let du=cos⁡(y)dydu = \cos{\left(y \right)} dy and substitute dudu:

        ∫eu du\int e^{u}\, du

        1. The integral of the exponential function is itself.

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

        Now substitute uu back in:

        esin⁡(y)e^{\sin{\left(y \right)}}

      So, the result is: exesin⁡(y)e^{x} e^{\sin{\left(y \right)}}

  2. Add the constant of integration:

    ex+sin⁡(y)+constante^{x + \sin{\left(y \right)}}+ \mathrm{constant}

The answer is:

ex+sin⁡(y)+constante^{x + \sin{\left(y \right)}}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                       
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 |  x + sin(y)                  x + sin(y)
 | E          *cos(y) dy = C + e          
 |                                        
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∫ex+sin⁡(y)cos⁡(y) dy=C+ex+sin⁡(y)\int e^{x + \sin{\left(y \right)}} \cos{\left(y \right)}\, dy = C + e^{x + \sin{\left(y \right)}}
Simplify
The answer [src] 
   x    x  sin(1)
- e  + e *e
−ex+exesin⁡(1)- e^{x} + e^{x} e^{\sin{\left(1 \right)}} 
=
= 
   x    x  sin(1)
- e  + e *e
−ex+exesin⁡(1)- e^{x} + e^{x} e^{\sin{\left(1 \right)}} 
-exp(x) + exp(x)*exp(sin(1))

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

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