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Identical expressions
e^(-2x)*sin(3x)
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e(-2x)*sin(3x)
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Similar expressions
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Solving integrals/ e^(-2x)*sin(3x)

Integral of e^(-2x)*sin(3x) dx

The solution

You have entered [src]

  1                  
  /                  
 |                   
 |   -2*x            
 |  e    *sin(3*x) dx
 |                   
/                    
0

$$\int\limits_{0}^{1} e^{- 2 x} \sin{\left(3 x \right)}\, dx$$

Simplify

Detail solution

Use integration by parts, noting that the integrand eventually repeats itself.
1. For the integrand $e^{- 2 x} \sin{\left(3 x \right)}$ :
  
  Let $u{\left(x \right)} = \sin{\left(3 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{- 2 x}$ .
  
  Then $\int e^{- 2 x} \sin{\left(3 x \right)}\, dx = - \int \left(- \frac{3 e^{- 2 x} \cos{\left(3 x \right)}}{2}\right)\, dx - \frac{e^{- 2 x} \sin{\left(3 x \right)}}{2}$ .
2. For the integrand $- \frac{3 e^{- 2 x} \cos{\left(3 x \right)}}{2}$ :
  
  Let $u{\left(x \right)} = - \frac{3 \cos{\left(3 x \right)}}{2}$ and let $\operatorname{dv}{\left(x \right)} = e^{- 2 x}$ .
  
  Then $\int e^{- 2 x} \sin{\left(3 x \right)}\, dx = \int \left(- \frac{9 e^{- 2 x} \sin{\left(3 x \right)}}{4}\right)\, dx - \frac{e^{- 2 x} \sin{\left(3 x \right)}}{2} - \frac{3 e^{- 2 x} \cos{\left(3 x \right)}}{4}$ .
3. Notice that the integrand has repeated itself, so move it to one side:
  
  $\frac{13 \int e^{- 2 x} \sin{\left(3 x \right)}\, dx}{4} = - \frac{e^{- 2 x} \sin{\left(3 x \right)}}{2} - \frac{3 e^{- 2 x} \cos{\left(3 x \right)}}{4}$
  
  Therefore,
  
  $\int e^{- 2 x} \sin{\left(3 x \right)}\, dx = - \frac{2 e^{- 2 x} \sin{\left(3 x \right)}}{13} - \frac{3 e^{- 2 x} \cos{\left(3 x \right)}}{13}$
Now simplify:

$- \frac{\left(2 \sin{\left(3 x \right)} + 3 \cos{\left(3 x \right)}\right) e^{- 2 x}}{13}$
Add the constant of integration:

$- \frac{\left(2 \sin{\left(3 x \right)} + 3 \cos{\left(3 x \right)}\right) e^{- 2 x}}{13}+ \mathrm{constant}$

The answer is:

$- \frac{\left(2 \sin{\left(3 x \right)} + 3 \cos{\left(3 x \right)}\right) e^{- 2 x}}{13}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                           
 |                                     -2*x      -2*x         
 |  -2*x                   3*cos(3*x)*e       2*e    *sin(3*x)
 | e    *sin(3*x) dx = C - ---------------- - ----------------
 |                                13                 13       
/

$${{e^ {- 2\,x }\,\left(-2\,\sin \left(3\,x\right)-3\,\cos \left(3\,x \right)\right)}\over{13}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

               -2      -2       
3    3*cos(3)*e     2*e  *sin(3)
-- - ------------ - ------------
13        13             13

$${{3}\over{13}}-{{e^ {- 2 }\,\left(2\,\sin 3+3\,\cos 3\right)}\over{ 13}}$$

               -2      -2       
3    3*cos(3)*e     2*e  *sin(3)
-- - ------------ - ------------
13        13             13

$$- \frac{2 \sin{\left(3 \right)}}{13 e^{2}} - \frac{3 \cos{\left(3 \right)}}{13 e^{2}} + \frac{3}{13}$$

Simplify

Numerical answer [src]

0.258749670174335

0.258749670174335

The graph

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

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Integral of e^(-2x)*sin(3x) dx

Limits of integration:

The graph:

Piecewise:

The solution