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e^(5x)×sin3x

Integral of e^(5x)×sin3x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                 
  /                 
 |                  
 |   5*x            
 |  e   *sin(3*x) dx
 |                  
/                   
0                   
$$\int\limits_{0}^{1} e^{5 x} \sin{\left(3 x \right)}\, dx$$
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]
  /                                                        
 |                                    5*x      5*x         
 |  5*x                   3*cos(3*x)*e      5*e   *sin(3*x)
 | e   *sin(3*x) dx = C - --------------- + ---------------
 |                               34                34      
/                                                          
$${{e^{5\,x}\,\left(5\,\sin \left(3\,x\right)-3\,\cos \left(3\,x \right)\right)}\over{34}}$$
The graph
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    
$$\frac{3}{34} + \frac{5 e^{5} \sin{\left(3 \right)}}{34} - \frac{3 e^{5} \cos{\left(3 \right)}}{34}$$
Numerical answer [src]
16.1324727284908
16.1324727284908
The graph
Integral of e^(5x)×sin3x dx

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