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Derivative of:
exp(2x)*sin(3x)
Identical expressions
exp(2x)*sin(3x)
exponent of (2x) multiply by sinus of (3x)
exp(2x)sin(3x)
exp2xsin3x
exp(2x)*sin(3x)dx

Solving integrals/ exp(2x)*sin(3x)

Integral of exp(2x)*sin(3x) dx

The solution

You have entered [src]

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

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

Integral(exp(2*x)*sin(3*x), (x, 0, 1))

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 $\int e^{2 x} \sin{\left(3 x \right)}\, dx = \frac{e^{2 x} \sin{\left(3 x \right)}}{2} - \int \frac{3 e^{2 x} \cos{\left(3 x \right)}}{2}\, dx$ .
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 $\int e^{2 x} \sin{\left(3 x \right)}\, dx = \frac{e^{2 x} \sin{\left(3 x \right)}}{2} - \frac{3 e^{2 x} \cos{\left(3 x \right)}}{4} + \int \left(- \frac{9 e^{2 x} \sin{\left(3 x \right)}}{4}\right)\, dx$ .
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 answer [src]

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

$${{2\,e^2\,\sin 3-3\,e^2\,\cos 3}\over{13}}+{{3}\over{13}}$$

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

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

Упростить

Numerical answer [src]

2.07929366132116

2.07929366132116

The graph

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

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Integral of exp(2x)*sin(3x) dx

Limits of integration:

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The solution