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Integral of e^(-2x)*sin(5x)
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Identical expressions
e^(-2x)*sin(5x)
e to the power of ( minus 2x) multiply by sinus of (5x)
e(-2x)*sin(5x)
e-2x*sin5x
e^(-2x)sin(5x)
e(-2x)sin(5x)
e-2xsin5x
e^-2xsin5x
e^(-2x)*sin(5x)dx
Similar expressions
e^(2x)*sin(5x)

Integral of e^(-2x)*sin(5x) dx

The solution

You have entered [src]

 oo                  
  /                  
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 |   -2*x            
 |  e    *sin(5*x) dx
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0

$$\int\limits_{0}^{\infty} e^{- 2 x} \sin{\left(5 x \right)}\, dx$$

Simplify

Detail solution

Use integration by parts, noting that the integrand eventually repeats itself.
1. For the integrand $e^{- 2 x} \sin{\left(5 x \right)}$ :
  
  Let $u{\left(x \right)} = \sin{\left(5 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{- 2 x}$ .
  
  Then $\int e^{- 2 x} \sin{\left(5 x \right)}\, dx = - \int \left(- \frac{5 e^{- 2 x} \cos{\left(5 x \right)}}{2}\right)\, dx - \frac{e^{- 2 x} \sin{\left(5 x \right)}}{2}$ .
2. For the integrand $- \frac{5 e^{- 2 x} \cos{\left(5 x \right)}}{2}$ :
  
  Let $u{\left(x \right)} = - \frac{5 \cos{\left(5 x \right)}}{2}$ and let $\operatorname{dv}{\left(x \right)} = e^{- 2 x}$ .
  
  Then $\int e^{- 2 x} \sin{\left(5 x \right)}\, dx = \int \left(- \frac{25 e^{- 2 x} \sin{\left(5 x \right)}}{4}\right)\, dx - \frac{e^{- 2 x} \sin{\left(5 x \right)}}{2} - \frac{5 e^{- 2 x} \cos{\left(5 x \right)}}{4}$ .
3. Notice that the integrand has repeated itself, so move it to one side:
  
  $\frac{29 \int e^{- 2 x} \sin{\left(5 x \right)}\, dx}{4} = - \frac{e^{- 2 x} \sin{\left(5 x \right)}}{2} - \frac{5 e^{- 2 x} \cos{\left(5 x \right)}}{4}$
  
  Therefore,
  
  $\int e^{- 2 x} \sin{\left(5 x \right)}\, dx = - \frac{2 e^{- 2 x} \sin{\left(5 x \right)}}{29} - \frac{5 e^{- 2 x} \cos{\left(5 x \right)}}{29}$
Now simplify:

$- \frac{\left(2 \sin{\left(5 x \right)} + 5 \cos{\left(5 x \right)}\right) e^{- 2 x}}{29}$
Add the constant of integration:

$- \frac{\left(2 \sin{\left(5 x \right)} + 5 \cos{\left(5 x \right)}\right) e^{- 2 x}}{29}+ \mathrm{constant}$

The answer is:

$- \frac{\left(2 \sin{\left(5 x \right)} + 5 \cos{\left(5 x \right)}\right) e^{- 2 x}}{29}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                           
 |                                     -2*x      -2*x         
 |  -2*x                   5*cos(5*x)*e       2*e    *sin(5*x)
 | e    *sin(5*x) dx = C - ---------------- - ----------------
 |                                29                 29       
/

$${{e^ {- 2\,x }\,\left(-2\,\sin \left(5\,x\right)-5\,\cos \left(5\,x \right)\right)}\over{29}}$$

Simplify

The answer [src]

5/29

$$\frac{5}{29}$$

5/29

$$\frac{5}{29}$$

Simplify

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

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Identical expressions

Similar expressions

Integral of e^(-2x)*sin(5x) dx

Limits of integration:

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Piecewise:

The solution