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e^(-x)*sin(x)
Solving integrals/ e^(-x)*sin(x)

Integral of e^(-x)*sin(x) dx

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The solution

You have entered [src] 
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∫01e−xsin⁡(x) dx\int\limits_{0}^{1} e^{- x} \sin{\left(x \right)}\, dx
Simplify
Detail solution

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

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

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

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

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

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

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

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

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

      Therefore,

      ∫e−xsin⁡(x) dx=−e−xsin⁡(x)2−e−xcos⁡(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:

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                           
 |                             -x    -x       
 |  -x                 cos(x)*e     e  *sin(x)
 | e  *sin(x) dx = C - ---------- - ----------
 |                         2            2     
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e−x (−sin⁡x−cos⁡x)2{{e^ {- x }\,\left(-\sin x-\cos x\right)}\over{2}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901.0-1.0
Plot the graph f(x)
The answer [src] 
            -1    -1       
1   cos(1)*e     e  *sin(1)
- - ---------- - ----------
2       2            2
12−e−1 (sin⁡1+cos⁡1)2{{1}\over{2}}-{{e^ {- 1 }\,\left(\sin 1+\cos 1\right)}\over{2}} 
=
= 
            -1    -1       
1   cos(1)*e     e  *sin(1)
- - ---------- - ----------
2       2            2
−sin⁡(1)2e−cos⁡(1)2e+12- \frac{\sin{\left(1 \right)}}{2 e} - \frac{\cos{\left(1 \right)}}{2 e} + \frac{1}{2}
Simplify
Numerical answer [src] 
0.245837007000237
0.245837007000237
The graph
Integral of e^(-x)*sin(x) dx

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

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