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Integral of d{x}:
Integral of e^(5x)×sin3x
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Identical expressions
e^(5x)×sin3x
e to the power of (5x)× sinus of 3x
e(5x)×sin3x
e5x×sin3x
e^5x×sin3x
e^(5x)×sin3xdx

Solving integrals/ e^(5x)×sin3x

Integral of e^(5x)×sin3x dx

The solution

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  1                 
  /                 
 |                  
 |   5*x            
 |  e   *sin(3*x) dx
 |                  
/                   
0

$$\int\limits_{0}^{1} e^{5 x} \sin{\left(3 x \right)}\, dx$$

Simplify

Detail solution

Use integration by parts, noting that the integrand eventually repeats itself.
1. For the integrand $e^{5 x} \sin{\left(3 x \right)}$ :
  
  Let $u{\left(x \right)} = \sin{\left(3 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{5 x}$ .
  
  Then $\int e^{5 x} \sin{\left(3 x \right)}\, dx = \frac{e^{5 x} \sin{\left(3 x \right)}}{5} - \int \frac{3 e^{5 x} \cos{\left(3 x \right)}}{5}\, dx$ .
2. For the integrand $\frac{3 e^{5 x} \cos{\left(3 x \right)}}{5}$ :
  
  Let $u{\left(x \right)} = \frac{3 \cos{\left(3 x \right)}}{5}$ and let $\operatorname{dv}{\left(x \right)} = e^{5 x}$ .
  
  Then $\int e^{5 x} \sin{\left(3 x \right)}\, dx = \frac{e^{5 x} \sin{\left(3 x \right)}}{5} - \frac{3 e^{5 x} \cos{\left(3 x \right)}}{25} + \int \left(- \frac{9 e^{5 x} \sin{\left(3 x \right)}}{25}\right)\, dx$ .
3. Notice that the integrand has repeated itself, so move it to one side:
  
  $\frac{34 \int e^{5 x} \sin{\left(3 x \right)}\, dx}{25} = \frac{e^{5 x} \sin{\left(3 x \right)}}{5} - \frac{3 e^{5 x} \cos{\left(3 x \right)}}{25}$
  
  Therefore,
  
  $\int e^{5 x} \sin{\left(3 x \right)}\, dx = \frac{5 e^{5 x} \sin{\left(3 x \right)}}{34} - \frac{3 e^{5 x} \cos{\left(3 x \right)}}{34}$
Now simplify:

$\frac{\left(5 \sin{\left(3 x \right)} - 3 \cos{\left(3 x \right)}\right) e^{5 x}}{34}$
Add the constant of integration:

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

The answer is:

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

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      
/

$${{e^{5\,x}\,\left(5\,\sin \left(3\,x\right)-3\,\cos \left(3\,x \right)\right)}\over{34}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

               5      5       
3    3*cos(3)*e    5*e *sin(3)
-- - ----------- + -----------
34        34            34

$${{5\,e^5\,\sin 3-3\,e^5\,\cos 3}\over{34}}+{{3}\over{34}}$$

               5      5       
3    3*cos(3)*e    5*e *sin(3)
-- - ----------- + -----------
34        34            34

$$\frac{3}{34} + \frac{5 e^{5} \sin{\left(3 \right)}}{34} - \frac{3 e^{5} \cos{\left(3 \right)}}{34}$$

Simplify

Numerical answer [src]

16.1324727284908

16.1324727284908

The graph

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

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Identical expressions

Integral of e^(5x)×sin3x dx

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The solution