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Similar expressions
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Integral of exp(2x)cos(x) dx

The solution

You have entered [src]

 pi               
 --               
 2                
  /               
 |                
 |   2*x          
 |  e   *cos(x) dx
 |                
/                 
0

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

Integral(exp(2*x)*cos(x), (x, 0, pi/2))

Detail solution

Use integration by parts, noting that the integrand eventually repeats itself.
1. For the integrand $e^{2 x} \cos{\left(x \right)}$ :
  
  Let $u{\left(x \right)} = \cos{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .
  
  Then $\int e^{2 x} \cos{\left(x \right)}\, dx = \frac{e^{2 x} \cos{\left(x \right)}}{2} - \int \left(- \frac{e^{2 x} \sin{\left(x \right)}}{2}\right)\, dx$ .
2. For the integrand $- \frac{e^{2 x} \sin{\left(x \right)}}{2}$ :
  
  Let $u{\left(x \right)} = - \frac{\sin{\left(x \right)}}{2}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .
  
  Then $\int e^{2 x} \cos{\left(x \right)}\, dx = \frac{e^{2 x} \sin{\left(x \right)}}{4} + \frac{e^{2 x} \cos{\left(x \right)}}{2} + \int \left(- \frac{e^{2 x} \cos{\left(x \right)}}{4}\right)\, dx$ .
3. Notice that the integrand has repeated itself, so move it to one side:
  
  $\frac{5 \int e^{2 x} \cos{\left(x \right)}\, dx}{4} = \frac{e^{2 x} \sin{\left(x \right)}}{4} + \frac{e^{2 x} \cos{\left(x \right)}}{2}$
  
  Therefore,
  
  $\int e^{2 x} \cos{\left(x \right)}\, dx = \frac{e^{2 x} \sin{\left(x \right)}}{5} + \frac{2 e^{2 x} \cos{\left(x \right)}}{5}$
Now simplify:

$\frac{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) e^{2 x}}{5}$
Add the constant of integration:

$\frac{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) e^{2 x}}{5}+ \mathrm{constant}$

The answer is:

$\frac{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) e^{2 x}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                
 |                       2*x                    2*x
 |  2*x                 e   *sin(x)   2*cos(x)*e   
 | e   *cos(x) dx = C + ----------- + -------------
 |                           5              5      
/

$$\int e^{2 x} \cos{\left(x \right)}\, dx = C + \frac{e^{2 x} \sin{\left(x \right)}}{5} + \frac{2 e^{2 x} \cos{\left(x \right)}}{5}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       pi
  2   e  
- - + ---
  5    5

$$- \frac{2}{5} + \frac{e^{\pi}}{5}$$

       pi
  2   e  
- - + ---
  5    5

$$- \frac{2}{5} + \frac{e^{\pi}}{5}$$

-2/5 + exp(pi)/5

Numerical answer [src]

4.22813852655585

4.22813852655585

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution