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e^(5x)*sin(x)
Solving integrals/ e^(5x)*sin(x)

Integral of e^(5x)*sin(x) dx

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The solution

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

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

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

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

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

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

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

      26∫e5xsin⁡(x) dx25=e5xsin⁡(x)5−e5xcos⁡(x)25\frac{26 \int e^{5 x} \sin{\left(x \right)}\, dx}{25} = \frac{e^{5 x} \sin{\left(x \right)}}{5} - \frac{e^{5 x} \cos{\left(x \right)}}{25}

      Therefore,

      ∫e5xsin⁡(x) dx=5e5xsin⁡(x)26−e5xcos⁡(x)26\int e^{5 x} \sin{\left(x \right)}\, dx = \frac{5 e^{5 x} \sin{\left(x \right)}}{26} - \frac{e^{5 x} \cos{\left(x \right)}}{26}

  2. Now simplify:

    (5sin⁡(x)−cos⁡(x))e5x26\frac{\left(5 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{5 x}}{26}

  3. Add the constant of integration:

    (5sin⁡(x)−cos⁡(x))e5x26+constant\frac{\left(5 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{5 x}}{26}+ \mathrm{constant}

The answer is:

(5sin⁡(x)−cos⁡(x))e5x26+constant\frac{\left(5 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{5 x}}{26}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                
 |                              5*x      5*x       
 |  5*x                 cos(x)*e      5*e   *sin(x)
 | e   *sin(x) dx = C - ----------- + -------------
 |                           26             26     
/
e5 x (5 sin⁡x−cos⁡x)26{{e^{5\,x}\,\left(5\,\sin x-\cos x\right)}\over{26}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90200-100
Plot the graph f(x)
The answer [src] 
             5      5       
1    cos(1)*e    5*e *sin(1)
-- - --------- + -----------
26       26           26
5 e5 sin⁡1−e5 cos⁡126+126{{5\,e^5\,\sin 1-e^5\,\cos 1}\over{26}}+{{1}\over{26}} 
=
= 
             5      5       
1    cos(1)*e    5*e *sin(1)
-- - --------- + -----------
26       26           26
−e5cos⁡(1)26+126+5e5sin⁡(1)26- \frac{e^{5} \cos{\left(1 \right)}}{26} + \frac{1}{26} + \frac{5 e^{5} \sin{\left(1 \right)}}{26}
Simplify
Numerical answer [src] 
20.9707255253148
20.9707255253148
The graph
Integral of e^(5x)*sin(x) dx

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

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