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e^(-2x)*sin(3x)
Solving integrals/ e^(-2x)*sin(3x)

Integral of e^(-2x)*sin(3x) dx

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The solution

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 |   -2*x            
 |  e    *sin(3*x) dx
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∫01e−2xsin⁡(3x) dx\int\limits_{0}^{1} e^{- 2 x} \sin{\left(3 x \right)}\, dx
Simplify
Detail solution

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

    1. For the integrand e−2xsin⁡(3x)e^{- 2 x} \sin{\left(3 x \right)}:

      Let u(x)=sin⁡(3x)u{\left(x \right)} = \sin{\left(3 x \right)} and let dv⁡(x)=e−2x\operatorname{dv}{\left(x \right)} = e^{- 2 x}.

      Then ∫e−2xsin⁡(3x) dx=−∫(−3e−2xcos⁡(3x)2) dx−e−2xsin⁡(3x)2\int e^{- 2 x} \sin{\left(3 x \right)}\, dx = - \int \left(- \frac{3 e^{- 2 x} \cos{\left(3 x \right)}}{2}\right)\, dx - \frac{e^{- 2 x} \sin{\left(3 x \right)}}{2}.

    2. For the integrand −3e−2xcos⁡(3x)2- \frac{3 e^{- 2 x} \cos{\left(3 x \right)}}{2}:

      Let u(x)=−3cos⁡(3x)2u{\left(x \right)} = - \frac{3 \cos{\left(3 x \right)}}{2} and let dv⁡(x)=e−2x\operatorname{dv}{\left(x \right)} = e^{- 2 x}.

      Then ∫e−2xsin⁡(3x) dx=∫(−9e−2xsin⁡(3x)4) dx−e−2xsin⁡(3x)2−3e−2xcos⁡(3x)4\int e^{- 2 x} \sin{\left(3 x \right)}\, dx = \int \left(- \frac{9 e^{- 2 x} \sin{\left(3 x \right)}}{4}\right)\, dx - \frac{e^{- 2 x} \sin{\left(3 x \right)}}{2} - \frac{3 e^{- 2 x} \cos{\left(3 x \right)}}{4}.

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

      13∫e−2xsin⁡(3x) dx4=−e−2xsin⁡(3x)2−3e−2xcos⁡(3x)4\frac{13 \int e^{- 2 x} \sin{\left(3 x \right)}\, dx}{4} = - \frac{e^{- 2 x} \sin{\left(3 x \right)}}{2} - \frac{3 e^{- 2 x} \cos{\left(3 x \right)}}{4}

      Therefore,

      ∫e−2xsin⁡(3x) dx=−2e−2xsin⁡(3x)13−3e−2xcos⁡(3x)13\int e^{- 2 x} \sin{\left(3 x \right)}\, dx = - \frac{2 e^{- 2 x} \sin{\left(3 x \right)}}{13} - \frac{3 e^{- 2 x} \cos{\left(3 x \right)}}{13}

  2. Now simplify:

    −(2sin⁡(3x)+3cos⁡(3x))e−2x13- \frac{\left(2 \sin{\left(3 x \right)} + 3 \cos{\left(3 x \right)}\right) e^{- 2 x}}{13}

  3. Add the constant of integration:

    −(2sin⁡(3x)+3cos⁡(3x))e−2x13+constant- \frac{\left(2 \sin{\left(3 x \right)} + 3 \cos{\left(3 x \right)}\right) e^{- 2 x}}{13}+ \mathrm{constant}

The answer is:

−(2sin⁡(3x)+3cos⁡(3x))e−2x13+constant- \frac{\left(2 \sin{\left(3 x \right)} + 3 \cos{\left(3 x \right)}\right) e^{- 2 x}}{13}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                           
 |                                     -2*x      -2*x         
 |  -2*x                   3*cos(3*x)*e       2*e    *sin(3*x)
 | e    *sin(3*x) dx = C - ---------------- - ----------------
 |                                13                 13       
/
e−2 x (−2 sin⁡(3 x)−3 cos⁡(3 x))13{{e^ {- 2\,x }\,\left(-2\,\sin \left(3\,x\right)-3\,\cos \left(3\,x \right)\right)}\over{13}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.5-0.5
Plot the graph f(x)
The answer [src] 
               -2      -2       
3    3*cos(3)*e     2*e  *sin(3)
-- - ------------ - ------------
13        13             13
313−e−2 (2 sin⁡3+3 cos⁡3)13{{3}\over{13}}-{{e^ {- 2 }\,\left(2\,\sin 3+3\,\cos 3\right)}\over{ 13}} 
=
= 
               -2      -2       
3    3*cos(3)*e     2*e  *sin(3)
-- - ------------ - ------------
13        13             13
−2sin⁡(3)13e2−3cos⁡(3)13e2+313- \frac{2 \sin{\left(3 \right)}}{13 e^{2}} - \frac{3 \cos{\left(3 \right)}}{13 e^{2}} + \frac{3}{13}
Simplify
Numerical answer [src] 
0.258749670174335
0.258749670174335
The graph
Integral of e^(-2x)*sin(3x) dx

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

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