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Derivative of:
e^(2x)cosx
Identical expressions
e^(2x)cosx
e to the power of (2x) co sinus of e of x
e(2x)cosx
e2xcosx
e^2xcosx
e^(2x)cosxdx

Solving integrals/ e^(2x)cosx

Integral of e^(2x)cosx dx

The solution

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

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

Integral(E^(2*x)*cos(x), (x, 0, 1))

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      
/

$${{e^{2\,x}\,\left(\sin x+2\,\cos x\right)}\over{5}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       2                    2
  2   e *sin(1)   2*cos(1)*e 
- - + --------- + -----------
  5       5            5

$${{e^2\,\sin 1+2\,e^2\,\cos 1}\over{5}}-{{2}\over{5}}$$

       2                    2
  2   e *sin(1)   2*cos(1)*e 
- - + --------- + -----------
  5       5            5

$$- \frac{2}{5} + \frac{e^{2} \sin{\left(1 \right)}}{5} + \frac{2 e^{2} \cos{\left(1 \right)}}{5}$$

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Numerical answer [src]

2.4404648818501

2.4404648818501

The graph

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

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Identical expressions

Integral of e^(2x)cosx dx

Limits of integration:

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Piecewise:

The solution