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Identical expressions
exp(two x)*cos(x/2)
exponent of (2x) multiply by co sinus of e of (x divide by 2)
exponent of (two x) multiply by co sinus of e of (x divide by 2)
exp(2x)cos(x/2)
exp2xcosx/2
exp(2x)*cos(x divide by 2)
exp(2x)*cos(x/2)dx

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

The solution

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  1               
  /               
 |                
 |   2*x    /x\   
 |  e   *cos|-| dx
 |          \2/   
 |                
/                 
0

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

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

Detail solution

Use integration by parts, noting that the integrand eventually repeats itself.
1. For the integrand $e^{2 x} \cos{\left(\frac{x}{2} \right)}$ :
  
  Let $u{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .
  
  Then $\int e^{2 x} \cos{\left(\frac{x}{2} \right)}\, dx = \frac{e^{2 x} \cos{\left(\frac{x}{2} \right)}}{2} - \int \left(- \frac{e^{2 x} \sin{\left(\frac{x}{2} \right)}}{4}\right)\, dx$ .
2. For the integrand $- \frac{e^{2 x} \sin{\left(\frac{x}{2} \right)}}{4}$ :
  
  Let $u{\left(x \right)} = - \frac{\sin{\left(\frac{x}{2} \right)}}{4}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .
  
  Then $\int e^{2 x} \cos{\left(\frac{x}{2} \right)}\, dx = \frac{e^{2 x} \sin{\left(\frac{x}{2} \right)}}{8} + \frac{e^{2 x} \cos{\left(\frac{x}{2} \right)}}{2} + \int \left(- \frac{e^{2 x} \cos{\left(\frac{x}{2} \right)}}{16}\right)\, dx$ .
3. Notice that the integrand has repeated itself, so move it to one side:
  
  $\frac{17 \int e^{2 x} \cos{\left(\frac{x}{2} \right)}\, dx}{16} = \frac{e^{2 x} \sin{\left(\frac{x}{2} \right)}}{8} + \frac{e^{2 x} \cos{\left(\frac{x}{2} \right)}}{2}$
  
  Therefore,
  
  $\int e^{2 x} \cos{\left(\frac{x}{2} \right)}\, dx = \frac{2 e^{2 x} \sin{\left(\frac{x}{2} \right)}}{17} + \frac{8 e^{2 x} \cos{\left(\frac{x}{2} \right)}}{17}$
Now simplify:

$\frac{2 \left(\sin{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{2} \right)}\right) e^{2 x}}{17}$
Add the constant of integration:

$\frac{2 \left(\sin{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{2} \right)}\right) e^{2 x}}{17}+ \mathrm{constant}$

The answer is:

\frac{2 \left(\sin{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{2} \right)}\right) e^{2 x}}{17}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                        2*x    /x\        /x\  2*x
 |                      2*e   *sin|-|   8*cos|-|*e   
 |  2*x    /x\                    \2/        \2/     
 | e   *cos|-| dx = C + ------------- + -------------
 |         \2/                17              17     
 |                                                   
/

\int e^{2 x} \cos{\left(\frac{x}{2} \right)}\, dx = C + \frac{2 e^{2 x} \sin{\left(\frac{x}{2} \right)}}{17} + \frac{8 e^{2 x} \cos{\left(\frac{x}{2} \right)}}{17}

Simplify

The graph

Plot the graph f(x)

The answer [src]

          2                        2
  8    2*e *sin(1/2)   8*cos(1/2)*e 
- -- + ------------- + -------------
  17         17              17

- \frac{8}{17} + \frac{2 e^{2} \sin{\left(\frac{1}{2} \right)}}{17} + \frac{8 e^{2} \cos{\left(\frac{1}{2} \right)}}{17}

          2                        2
  8    2*e *sin(1/2)   8*cos(1/2)*e 
- -- + ------------- + -------------
  17         17              17

- \frac{8}{17} + \frac{2 e^{2} \sin{\left(\frac{1}{2} \right)}}{17} + \frac{8 e^{2} \cos{\left(\frac{1}{2} \right)}}{17}

-8/17 + 2*exp(2)*sin(1/2)/17 + 8*cos(1/2)*exp(2)/17

Numerical answer [src]

2.99770933235429

2.99770933235429

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution