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Solving integrals/ (exp(3x))*cos(2x)

Integral of (exp(3x))*cos(2x) dx

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The solution

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 |  e   *cos(2*x) dx
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∫01e3xcos⁡(2x) dx\int\limits_{0}^{1} e^{3 x} \cos{\left(2 x \right)}\, dx 
Integral(exp(3*x)*cos(2*x), (x, 0, 1))
Detail solution

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

    1. For the integrand e3xcos⁡(2x)e^{3 x} \cos{\left(2 x \right)}:

      Let u(x)=cos⁡(2x)u{\left(x \right)} = \cos{\left(2 x \right)} and let dv⁡(x)=e3x\operatorname{dv}{\left(x \right)} = e^{3 x}.

      Then ∫e3xcos⁡(2x) dx=e3xcos⁡(2x)3−∫(−2e3xsin⁡(2x)3) dx\int e^{3 x} \cos{\left(2 x \right)}\, dx = \frac{e^{3 x} \cos{\left(2 x \right)}}{3} - \int \left(- \frac{2 e^{3 x} \sin{\left(2 x \right)}}{3}\right)\, dx.

    2. For the integrand −2e3xsin⁡(2x)3- \frac{2 e^{3 x} \sin{\left(2 x \right)}}{3}:

      Let u(x)=−2sin⁡(2x)3u{\left(x \right)} = - \frac{2 \sin{\left(2 x \right)}}{3} and let dv⁡(x)=e3x\operatorname{dv}{\left(x \right)} = e^{3 x}.

      Then ∫e3xcos⁡(2x) dx=2e3xsin⁡(2x)9+e3xcos⁡(2x)3+∫(−4e3xcos⁡(2x)9) dx\int e^{3 x} \cos{\left(2 x \right)}\, dx = \frac{2 e^{3 x} \sin{\left(2 x \right)}}{9} + \frac{e^{3 x} \cos{\left(2 x \right)}}{3} + \int \left(- \frac{4 e^{3 x} \cos{\left(2 x \right)}}{9}\right)\, dx.

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

      13∫e3xcos⁡(2x) dx9=2e3xsin⁡(2x)9+e3xcos⁡(2x)3\frac{13 \int e^{3 x} \cos{\left(2 x \right)}\, dx}{9} = \frac{2 e^{3 x} \sin{\left(2 x \right)}}{9} + \frac{e^{3 x} \cos{\left(2 x \right)}}{3}

      Therefore,

      ∫e3xcos⁡(2x) dx=2e3xsin⁡(2x)13+3e3xcos⁡(2x)13\int e^{3 x} \cos{\left(2 x \right)}\, dx = \frac{2 e^{3 x} \sin{\left(2 x \right)}}{13} + \frac{3 e^{3 x} \cos{\left(2 x \right)}}{13}

  2. Now simplify:

    (2sin⁡(2x)+3cos⁡(2x))e3x13\frac{\left(2 \sin{\left(2 x \right)} + 3 \cos{\left(2 x \right)}\right) e^{3 x}}{13}

  3. Add the constant of integration:

    (2sin⁡(2x)+3cos⁡(2x))e3x13+constant\frac{\left(2 \sin{\left(2 x \right)} + 3 \cos{\left(2 x \right)}\right) e^{3 x}}{13}+ \mathrm{constant}

The answer is:

(2sin⁡(2x)+3cos⁡(2x))e3x13+constant\frac{\left(2 \sin{\left(2 x \right)} + 3 \cos{\left(2 x \right)}\right) e^{3 x}}{13}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                        
 |                           3*x                        3*x
 |  3*x                   2*e   *sin(2*x)   3*cos(2*x)*e   
 | e   *cos(2*x) dx = C + --------------- + ---------------
 |                               13                13      
/
∫e3xcos⁡(2x) dx=C+2e3xsin⁡(2x)13+3e3xcos⁡(2x)13\int e^{3 x} \cos{\left(2 x \right)}\, dx = C + \frac{2 e^{3 x} \sin{\left(2 x \right)}}{13} + \frac{3 e^{3 x} \cos{\left(2 x \right)}}{13}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-1010
Plot the graph f(x)
The answer [src] 
          3                    3
  3    2*e *sin(2)   3*cos(2)*e 
- -- + ----------- + -----------
  13        13            13
3e3cos⁡(2)13−313+2e3sin⁡(2)13\frac{3 e^{3} \cos{\left(2 \right)}}{13} - \frac{3}{13} + \frac{2 e^{3} \sin{\left(2 \right)}}{13} 
=
= 
          3                    3
  3    2*e *sin(2)   3*cos(2)*e 
- -- + ----------- + -----------
  13        13            13
3e3cos⁡(2)13−313+2e3sin⁡(2)13\frac{3 e^{3} \cos{\left(2 \right)}}{13} - \frac{3}{13} + \frac{2 e^{3} \sin{\left(2 \right)}}{13} 
-3/13 + 2*exp(3)*sin(2)/13 + 3*cos(2)*exp(3)/13
Numerical answer [src] 
0.650142779117494
0.650142779117494

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

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