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

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

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

  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)=sin⁡2(x)u{\left(x \right)} = \sin^{2}{\left(x \right)} and let dv⁡(x)=ex\operatorname{dv}{\left(x \right)} = e^{x}.

    Then du⁡(x)=2sin⁡(x)cos⁡(x)\operatorname{du}{\left(x \right)} = 2 \sin{\left(x \right)} \cos{\left(x \right)}.

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

    1. The integral of the exponential function is itself.

      ∫ex dx=ex\int e^{x}\, dx = e^{x}

    Now evaluate the sub-integral.

  2. 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)=2sin⁡(x)cos⁡(x)u{\left(x \right)} = 2 \sin{\left(x \right)} \cos{\left(x \right)} and let dv⁡(x)=ex\operatorname{dv}{\left(x \right)} = e^{x}.

    Then du⁡(x)=−2sin⁡2(x)+2cos⁡2(x)\operatorname{du}{\left(x \right)} = - 2 \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}.

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

    1. The integral of the exponential function is itself.

      ∫ex dx=ex\int e^{x}\, dx = e^{x}

    Now evaluate the sub-integral.

  3. Rewrite the integrand:

    (−2sin⁡2(x)+2cos⁡2(x))ex=−2exsin⁡2(x)+2excos⁡2(x)\left(- 2 \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right) e^{x} = - 2 e^{x} \sin^{2}{\left(x \right)} + 2 e^{x} \cos^{2}{\left(x \right)}

  4. Integrate term-by-term:

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

      ∫(−2exsin⁡2(x)) dx=−2∫exsin⁡2(x) dx\int \left(- 2 e^{x} \sin^{2}{\left(x \right)}\right)\, dx = - 2 \int e^{x} \sin^{2}{\left(x \right)}\, dx

      1. Don't know the steps in finding this integral.

        But the integral is

        3exsin⁡2(x)5−2exsin⁡(x)cos⁡(x)5+2excos⁡2(x)5\frac{3 e^{x} \sin^{2}{\left(x \right)}}{5} - \frac{2 e^{x} \sin{\left(x \right)} \cos{\left(x \right)}}{5} + \frac{2 e^{x} \cos^{2}{\left(x \right)}}{5}

      So, the result is: −6exsin⁡2(x)5+4exsin⁡(x)cos⁡(x)5−4excos⁡2(x)5- \frac{6 e^{x} \sin^{2}{\left(x \right)}}{5} + \frac{4 e^{x} \sin{\left(x \right)} \cos{\left(x \right)}}{5} - \frac{4 e^{x} \cos^{2}{\left(x \right)}}{5}

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

      ∫2excos⁡2(x) dx=2∫excos⁡2(x) dx\int 2 e^{x} \cos^{2}{\left(x \right)}\, dx = 2 \int e^{x} \cos^{2}{\left(x \right)}\, dx

      1. Don't know the steps in finding this integral.

        But the integral is

        2exsin⁡2(x)5+2exsin⁡(x)cos⁡(x)5+3excos⁡2(x)5\frac{2 e^{x} \sin^{2}{\left(x \right)}}{5} + \frac{2 e^{x} \sin{\left(x \right)} \cos{\left(x \right)}}{5} + \frac{3 e^{x} \cos^{2}{\left(x \right)}}{5}

      So, the result is: 4exsin⁡2(x)5+4exsin⁡(x)cos⁡(x)5+6excos⁡2(x)5\frac{4 e^{x} \sin^{2}{\left(x \right)}}{5} + \frac{4 e^{x} \sin{\left(x \right)} \cos{\left(x \right)}}{5} + \frac{6 e^{x} \cos^{2}{\left(x \right)}}{5}

    The result is: −2exsin⁡2(x)5+8exsin⁡(x)cos⁡(x)5+2excos⁡2(x)5- \frac{2 e^{x} \sin^{2}{\left(x \right)}}{5} + \frac{8 e^{x} \sin{\left(x \right)} \cos{\left(x \right)}}{5} + \frac{2 e^{x} \cos^{2}{\left(x \right)}}{5}

  5. Now simplify:

    (−2sin⁡(2x)−cos⁡(2x)+5)ex10\frac{\left(- 2 \sin{\left(2 x \right)} - \cos{\left(2 x \right)} + 5\right) e^{x}}{10}

  6. Add the constant of integration:

    (−2sin⁡(2x)−cos⁡(2x)+5)ex10+constant\frac{\left(- 2 \sin{\left(2 x \right)} - \cos{\left(2 x \right)} + 5\right) e^{x}}{10}+ \mathrm{constant}

The answer is:

(−2sin⁡(2x)−cos⁡(2x)+5)ex10+constant\frac{\left(- 2 \sin{\left(2 x \right)} - \cos{\left(2 x \right)} + 5\right) e^{x}}{10}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                          
 |                                2     x        2     x             x       
 |  x                        2*cos (x)*e    3*sin (x)*e    2*cos(x)*e *sin(x)
 | e *sin(x)*sin(x) dx = C + ------------ + ------------ - ------------------
 |                                5              5                 5         
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−2 ex sin⁡(2 x)+ex cos⁡(2 x)−5 ex10-{{2\,e^{x}\,\sin \left(2\,x\right)+e^{x}\,\cos \left(2\,x\right)-5 \,e^{x}}\over{10}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
             2             2                       
  2   2*e*cos (1)   3*e*sin (1)   2*e*cos(1)*sin(1)
- - + ----------- + ----------- - -----------------
  5        5             5                5
−2 e sin⁡2+e cos⁡2−5 e10−25-{{2\,e\,\sin 2+e\,\cos 2-5\,e}\over{10}}-{{2}\over{5}} 
=
= 
             2             2                       
  2   2*e*cos (1)   3*e*sin (1)   2*e*cos(1)*sin(1)
- - + ----------- + ----------- - -----------------
  5        5             5                5
−2esin⁡(1)cos⁡(1)5−25+2ecos⁡2(1)5+3esin⁡2(1)5- \frac{2 e \sin{\left(1 \right)} \cos{\left(1 \right)}}{5} - \frac{2}{5} + \frac{2 e \cos^{2}{\left(1 \right)}}{5} + \frac{3 e \sin^{2}{\left(1 \right)}}{5}
Simplify
Numerical answer [src] 
0.57791601820424
0.57791601820424
The graph
Integral of exp(x)sin(x)sin(x) dx

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

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