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

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

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The solution

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

  1. Use integration by parts, noting that the integrand eventually repeats itself.

    1. For the integrand exsin⁡(x)e^{x} \sin{\left(x \right)}:

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

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

    2. For the integrand excos⁡(x)e^{x} \cos{\left(x \right)}:

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

      Then ∫exsin⁡(x) dx=exsin⁡(x)−excos⁡(x)+∫(−exsin⁡(x)) dx\int e^{x} \sin{\left(x \right)}\, dx = e^{x} \sin{\left(x \right)} - e^{x} \cos{\left(x \right)} + \int \left(- e^{x} \sin{\left(x \right)}\right)\, dx.

    3. Notice that the integrand has repeated itself, so move it to one side:

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

      Therefore,

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

  2. Now simplify:

    −2excos⁡(x+π4)2- \frac{\sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}}{2}

  3. Add the constant of integration:

    −2excos⁡(x+π4)2+constant- \frac{\sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}

The answer is:

−2excos⁡(x+π4)2+constant- \frac{\sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                        
 |                     x                  x
 |  x                 e *sin(x)   cos(x)*e 
 | e *sin(x) dx = C + --------- - ---------
 |                        2           2    
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ex (sin⁡x−cos⁡x)2{{e^{x}\,\left(\sin x-\cos x\right)}\over{2}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.905-5
Plot the graph f(x)
The answer [src] 
1   e*sin(1)   e*cos(1)
- + -------- - --------
2      2          2
e sin⁡1−e cos⁡12+12{{e\,\sin 1-e\,\cos 1}\over{2}}+{{1}\over{2}} 
=
= 
1   e*sin(1)   e*cos(1)
- + -------- - --------
2      2          2
−ecos⁡(1)2+12+esin⁡(1)2- \frac{e \cos{\left(1 \right)}}{2} + \frac{1}{2} + \frac{e \sin{\left(1 \right)}}{2}
Simplify
Numerical answer [src] 
0.909330673631479
0.909330673631479
The graph
Integral of exp(x)*sin(x) dx

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

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