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e^(-2x)*sin(3x)

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                  
  /                  
 |                   
 |   -2*x            
 |  e    *sin(3*x) dx
 |                   
/                    
0                    
$$\int\limits_{0}^{1} e^{- 2 x} \sin{\left(3 x \right)}\, dx$$
Detail solution
  1. Use integration by parts, noting that the integrand eventually repeats itself.

    1. For the integrand :

      Let and let .

      Then .

    2. For the integrand :

      Let and let .

      Then .

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

      Therefore,

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

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       
/                                                             
$${{e^ {- 2\,x }\,\left(-2\,\sin \left(3\,x\right)-3\,\cos \left(3\,x \right)\right)}\over{13}}$$
The graph
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     
$$- \frac{2 \sin{\left(3 \right)}}{13 e^{2}} - \frac{3 \cos{\left(3 \right)}}{13 e^{2}} + \frac{3}{13}$$
Numerical answer [src]
0.258749670174335
0.258749670174335
The graph
Integral of e^(-2x)*sin(3x) dx

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