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cos(x)*exp(sin(x))*sin(x)
Solving integrals/ cos(x)*exp(sin(x))*sin(x)

Integral of cos(x)*exp(sin(x))*sin(x) dx

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The solution

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 |          sin(x)          
 |  cos(x)*e      *sin(x) dx
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∫01esin⁡(x)cos⁡(x)sin⁡(x) dx\int\limits_{0}^{1} e^{\sin{\left(x \right)}} \cos{\left(x \right)} \sin{\left(x \right)}\, dx 
Integral((cos(x)*exp(sin(x)))*sin(x), (x, 0, 1))
Detail solution

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

    Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

    ∫ueu du\int u e^{u}\, du

    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(u)=uu{\left(u \right)} = u and let dv⁡(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

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

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

      1. The integral of the exponential function is itself.

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

      Now evaluate the sub-integral.

    2. The integral of the exponential function is itself.

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

    Now substitute uu back in:

    esin⁡(x)sin⁡(x)−esin⁡(x)e^{\sin{\left(x \right)}} \sin{\left(x \right)} - e^{\sin{\left(x \right)}}

  2. Now simplify:

    (sin⁡(x)−1)esin⁡(x)\left(\sin{\left(x \right)} - 1\right) e^{\sin{\left(x \right)}}

  3. Add the constant of integration:

    (sin⁡(x)−1)esin⁡(x)+constant\left(\sin{\left(x \right)} - 1\right) e^{\sin{\left(x \right)}}+ \mathrm{constant}

The answer is:

(sin⁡(x)−1)esin⁡(x)+constant\left(\sin{\left(x \right)} - 1\right) e^{\sin{\left(x \right)}}+ \mathrm{constant}

The answer (Indefinite) [src] 
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 |         sin(x)                  sin(x)    sin(x)       
 | cos(x)*e      *sin(x) dx = C - e       + e      *sin(x)
 |                                                        
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∫esin⁡(x)cos⁡(x)sin⁡(x) dx=C+esin⁡(x)sin⁡(x)−esin⁡(x)\int e^{\sin{\left(x \right)}} \cos{\left(x \right)} \sin{\left(x \right)}\, dx = C + e^{\sin{\left(x \right)}} \sin{\left(x \right)} - e^{\sin{\left(x \right)}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902.5-2.5
Plot the graph f(x)
The answer [src] 
     sin(1)    sin(1)       
1 - e       + e      *sin(1)
−esin⁡(1)+1+esin⁡(1)sin⁡(1)- e^{\sin{\left(1 \right)}} + 1 + e^{\sin{\left(1 \right)}} \sin{\left(1 \right)} 
=
= 
     sin(1)    sin(1)       
1 - e       + e      *sin(1)
−esin⁡(1)+1+esin⁡(1)sin⁡(1)- e^{\sin{\left(1 \right)}} + 1 + e^{\sin{\left(1 \right)}} \sin{\left(1 \right)} 
1 - exp(sin(1)) + exp(sin(1))*sin(1)
Numerical answer [src] 
0.632248064512331
0.632248064512331
The graph
Integral of cos(x)*exp(sin(x))*sin(x) dx

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

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