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

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

The solution

You have entered [src]

  1               
  /               
 |                
 |   5*x          
 |  e   *sin(x) dx
 |                
/                 
0

$$\int\limits_{0}^{1} e^{5 x} \sin{\left(x \right)}\, dx$$

Simplify

Detail solution

Use integration by parts, noting that the integrand eventually repeats itself.
1. For the integrand $e^{5 x} \sin{\left(x \right)}$ :
  
  Let $u{\left(x \right)} = \sin{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{5 x}$ .
  
  Then $\int e^{5 x} \sin{\left(x \right)}\, dx = \frac{e^{5 x} \sin{\left(x \right)}}{5} - \int \frac{e^{5 x} \cos{\left(x \right)}}{5}\, dx$ .
2. For the integrand $\frac{e^{5 x} \cos{\left(x \right)}}{5}$ :
  
  Let $u{\left(x \right)} = \frac{\cos{\left(x \right)}}{5}$ and let $\operatorname{dv}{\left(x \right)} = e^{5 x}$ .
  
  Then $\int e^{5 x} \sin{\left(x \right)}\, dx = \frac{e^{5 x} \sin{\left(x \right)}}{5} - \frac{e^{5 x} \cos{\left(x \right)}}{25} + \int \left(- \frac{e^{5 x} \sin{\left(x \right)}}{25}\right)\, dx$ .
3. Notice that the integrand has repeated itself, so move it to one side:
  
  $\frac{26 \int e^{5 x} \sin{\left(x \right)}\, dx}{25} = \frac{e^{5 x} \sin{\left(x \right)}}{5} - \frac{e^{5 x} \cos{\left(x \right)}}{25}$
  
  Therefore,
  
  $\int e^{5 x} \sin{\left(x \right)}\, dx = \frac{5 e^{5 x} \sin{\left(x \right)}}{26} - \frac{e^{5 x} \cos{\left(x \right)}}{26}$
Now simplify:

$\frac{\left(5 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{5 x}}{26}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                
 |                              5*x      5*x       
 |  5*x                 cos(x)*e      5*e   *sin(x)
 | e   *sin(x) dx = C - ----------- + -------------
 |                           26             26     
/

$${{e^{5\,x}\,\left(5\,\sin x-\cos x\right)}\over{26}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

             5      5       
1    cos(1)*e    5*e *sin(1)
-- - --------- + -----------
26       26           26

$${{5\,e^5\,\sin 1-e^5\,\cos 1}\over{26}}+{{1}\over{26}}$$

             5      5       
1    cos(1)*e    5*e *sin(1)
-- - --------- + -----------
26       26           26

$$- \frac{e^{5} \cos{\left(1 \right)}}{26} + \frac{1}{26} + \frac{5 e^{5} \sin{\left(1 \right)}}{26}$$

Simplify

Numerical answer [src]

20.9707255253148

20.9707255253148

The graph

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

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

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The solution