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Integral of d{x}:
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Integral of dx/(1+x^2)
Integral of log(x)^3/x
Integral of x^2*e^(x^3)*dx
Identical expressions
(exp(3x))*cos(2x)
( exponent of (3x)) multiply by co sinus of e of (2x)
(exp(3x))cos(2x)
exp3xcos2x
(exp(3x))*cos(2x)dx

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

The solution

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

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

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

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\int e^{3 x} \cos{\left(2 x \right)}\, dx = C + \frac{2 e^{3 x} \sin{\left(2 x \right)}}{13} + \frac{3 e^{3 x} \cos{\left(2 x \right)}}{13}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

-3/13 + 2*exp(3)*sin(2)/13 + 3*cos(2)*exp(3)/13

Numerical answer [src]

0.650142779117494

0.650142779117494

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

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Identical expressions

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

Limits of integration:

The graph:

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The solution