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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  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
  1. Use integration by parts, noting that the integrand eventually repeats itself.

    1. For the integrand :

      Let and let .

      Then .

    2. For the integrand :

      Let and let .

      Then .

    3. Notice that the integrand has repeated itself, so move it to one side:

      Therefore,

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

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     
 |                                                   
/                                                    
$$\int e^{2 x} \cos{\left(\frac{x}{2} \right)}\, dx = C + \frac{2 e^{2 x} \sin{\left(\frac{x}{2} \right)}}{17} + \frac{8 e^{2 x} \cos{\left(\frac{x}{2} \right)}}{17}$$
The graph
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.