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e^(5x)×sin3x
Solving integrals/ e^(5x)×sin3x

Integral of e^(5x)×sin3x dx

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The solution

You have entered [src] 
  1                 
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 |   5*x            
 |  e   *sin(3*x) dx
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0
∫01e5xsin⁡(3x) dx\int\limits_{0}^{1} e^{5 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 e5xsin⁡(3x)e^{5 x} \sin{\left(3 x \right)}:

      Let u(x)=sin⁡(3x)u{\left(x \right)} = \sin{\left(3 x \right)} and let dv⁡(x)=e5x\operatorname{dv}{\left(x \right)} = e^{5 x}.

      Then ∫e5xsin⁡(3x) dx=e5xsin⁡(3x)5−∫3e5xcos⁡(3x)5 dx\int e^{5 x} \sin{\left(3 x \right)}\, dx = \frac{e^{5 x} \sin{\left(3 x \right)}}{5} - \int \frac{3 e^{5 x} \cos{\left(3 x \right)}}{5}\, dx.

    2. For the integrand 3e5xcos⁡(3x)5\frac{3 e^{5 x} \cos{\left(3 x \right)}}{5}:

      Let u(x)=3cos⁡(3x)5u{\left(x \right)} = \frac{3 \cos{\left(3 x \right)}}{5} and let dv⁡(x)=e5x\operatorname{dv}{\left(x \right)} = e^{5 x}.

      Then ∫e5xsin⁡(3x) dx=e5xsin⁡(3x)5−3e5xcos⁡(3x)25+∫(−9e5xsin⁡(3x)25) dx\int e^{5 x} \sin{\left(3 x \right)}\, dx = \frac{e^{5 x} \sin{\left(3 x \right)}}{5} - \frac{3 e^{5 x} \cos{\left(3 x \right)}}{25} + \int \left(- \frac{9 e^{5 x} \sin{\left(3 x \right)}}{25}\right)\, dx.

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

      34∫e5xsin⁡(3x) dx25=e5xsin⁡(3x)5−3e5xcos⁡(3x)25\frac{34 \int e^{5 x} \sin{\left(3 x \right)}\, dx}{25} = \frac{e^{5 x} \sin{\left(3 x \right)}}{5} - \frac{3 e^{5 x} \cos{\left(3 x \right)}}{25}

      Therefore,

      ∫e5xsin⁡(3x) dx=5e5xsin⁡(3x)34−3e5xcos⁡(3x)34\int e^{5 x} \sin{\left(3 x \right)}\, dx = \frac{5 e^{5 x} \sin{\left(3 x \right)}}{34} - \frac{3 e^{5 x} \cos{\left(3 x \right)}}{34}

  2. Now simplify:

    (5sin⁡(3x)−3cos⁡(3x))e5x34\frac{\left(5 \sin{\left(3 x \right)} - 3 \cos{\left(3 x \right)}\right) e^{5 x}}{34}

  3. Add the constant of integration:

    (5sin⁡(3x)−3cos⁡(3x))e5x34+constant\frac{\left(5 \sin{\left(3 x \right)} - 3 \cos{\left(3 x \right)}\right) e^{5 x}}{34}+ \mathrm{constant}

The answer is:

(5sin⁡(3x)−3cos⁡(3x))e5x34+constant\frac{\left(5 \sin{\left(3 x \right)} - 3 \cos{\left(3 x \right)}\right) e^{5 x}}{34}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                        
 |                                    5*x      5*x         
 |  5*x                   3*cos(3*x)*e      5*e   *sin(3*x)
 | e   *sin(3*x) dx = C - --------------- + ---------------
 |                               34                34      
/
e5 x (5 sin⁡(3 x)−3 cos⁡(3 x))34{{e^{5\,x}\,\left(5\,\sin \left(3\,x\right)-3\,\cos \left(3\,x \right)\right)}\over{34}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-5050
Plot the graph f(x)
The answer [src] 
               5      5       
3    3*cos(3)*e    5*e *sin(3)
-- - ----------- + -----------
34        34            34
5 e5 sin⁡3−3 e5 cos⁡334+334{{5\,e^5\,\sin 3-3\,e^5\,\cos 3}\over{34}}+{{3}\over{34}} 
=
= 
               5      5       
3    3*cos(3)*e    5*e *sin(3)
-- - ----------- + -----------
34        34            34
334+5e5sin⁡(3)34−3e5cos⁡(3)34\frac{3}{34} + \frac{5 e^{5} \sin{\left(3 \right)}}{34} - \frac{3 e^{5} \cos{\left(3 \right)}}{34}
Simplify
Numerical answer [src] 
16.1324727284908
16.1324727284908
The graph
Integral of e^(5x)×sin3x dx

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

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