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

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

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 |  cos(x)*e    dx
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∫01e2xcos⁡(x) dx\int\limits_{0}^{1} e^{2 x} \cos{\left(x \right)}\, dx 
Integral(cos(x)*exp(2*x), (x, 0, 1))
Detail solution

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

    1. For the integrand e2xcos⁡(x)e^{2 x} \cos{\left(x \right)}:

      Let u(x)=cos⁡(x)u{\left(x \right)} = \cos{\left(x \right)} and let dv⁡(x)=e2x\operatorname{dv}{\left(x \right)} = e^{2 x}.

      Then ∫e2xcos⁡(x) dx=e2xcos⁡(x)2−∫(−e2xsin⁡(x)2) dx\int e^{2 x} \cos{\left(x \right)}\, dx = \frac{e^{2 x} \cos{\left(x \right)}}{2} - \int \left(- \frac{e^{2 x} \sin{\left(x \right)}}{2}\right)\, dx.

    2. For the integrand −e2xsin⁡(x)2- \frac{e^{2 x} \sin{\left(x \right)}}{2}:

      Let u(x)=−sin⁡(x)2u{\left(x \right)} = - \frac{\sin{\left(x \right)}}{2} and let dv⁡(x)=e2x\operatorname{dv}{\left(x \right)} = e^{2 x}.

      Then ∫e2xcos⁡(x) dx=e2xsin⁡(x)4+e2xcos⁡(x)2+∫(−e2xcos⁡(x)4) dx\int e^{2 x} \cos{\left(x \right)}\, dx = \frac{e^{2 x} \sin{\left(x \right)}}{4} + \frac{e^{2 x} \cos{\left(x \right)}}{2} + \int \left(- \frac{e^{2 x} \cos{\left(x \right)}}{4}\right)\, dx.

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

      5∫e2xcos⁡(x) dx4=e2xsin⁡(x)4+e2xcos⁡(x)2\frac{5 \int e^{2 x} \cos{\left(x \right)}\, dx}{4} = \frac{e^{2 x} \sin{\left(x \right)}}{4} + \frac{e^{2 x} \cos{\left(x \right)}}{2}

      Therefore,

      ∫e2xcos⁡(x) dx=e2xsin⁡(x)5+2e2xcos⁡(x)5\int e^{2 x} \cos{\left(x \right)}\, dx = \frac{e^{2 x} \sin{\left(x \right)}}{5} + \frac{2 e^{2 x} \cos{\left(x \right)}}{5}

  2. Now simplify:

    (sin⁡(x)+2cos⁡(x))e2x5\frac{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) e^{2 x}}{5}

  3. Add the constant of integration:

    (sin⁡(x)+2cos⁡(x))e2x5+constant\frac{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) e^{2 x}}{5}+ \mathrm{constant}

The answer is:

(sin⁡(x)+2cos⁡(x))e2x5+constant\frac{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) e^{2 x}}{5}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                
 |                       2*x                    2*x
 |         2*x          e   *sin(x)   2*cos(x)*e   
 | cos(x)*e    dx = C + ----------- + -------------
 |                           5              5      
/
∫e2xcos⁡(x) dx=C+e2xsin⁡(x)5+2e2xcos⁡(x)5\int e^{2 x} \cos{\left(x \right)}\, dx = C + \frac{e^{2 x} \sin{\left(x \right)}}{5} + \frac{2 e^{2 x} \cos{\left(x \right)}}{5}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9005
Plot the graph f(x)
The answer [src] 
       2                    2
  2   e *sin(1)   2*cos(1)*e 
- - + --------- + -----------
  5       5            5
−25+e2sin⁡(1)5+2e2cos⁡(1)5- \frac{2}{5} + \frac{e^{2} \sin{\left(1 \right)}}{5} + \frac{2 e^{2} \cos{\left(1 \right)}}{5} 
=
= 
       2                    2
  2   e *sin(1)   2*cos(1)*e 
- - + --------- + -----------
  5       5            5
−25+e2sin⁡(1)5+2e2cos⁡(1)5- \frac{2}{5} + \frac{e^{2} \sin{\left(1 \right)}}{5} + \frac{2 e^{2} \cos{\left(1 \right)}}{5} 
-2/5 + exp(2)*sin(1)/5 + 2*cos(1)*exp(2)/5
Numerical answer [src] 
2.4404648818501
2.4404648818501
The graph
Integral of cos(x)*exp(2*x) dx

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

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