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Solving integrals/ (e^(-x))*(sin(x/2))

Integral of (e^(-x))*(sin(x/2)) dx

The solution

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

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

Integral(sin(x/2)/E^x, (x, 0, 1))

Detail solution

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

$- \frac{\left(4 \sin{\left(\frac{x}{2} \right)} + 2 \cos{\left(\frac{x}{2} \right)}\right) e^{- x}}{5}$
Add the constant of integration:

$- \frac{\left(4 \sin{\left(\frac{x}{2} \right)} + 2 \cos{\left(\frac{x}{2} \right)}\right) e^{- x}}{5}+ \mathrm{constant}$

The answer is:

$- \frac{\left(4 \sin{\left(\frac{x}{2} \right)} + 2 \cos{\left(\frac{x}{2} \right)}\right) e^{- x}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                       -x    /x\        /x\  -x
 |                     4*e  *sin|-|   2*cos|-|*e  
 |  -x    /x\                   \2/        \2/    
 | e  *sin|-| dx = C - ------------ - ------------
 |        \2/               5              5      
 |                                                
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

       -1                        -1
2   4*e  *sin(1/2)   2*cos(1/2)*e  
- - -------------- - --------------
5         5                5

$${{2}\over{5}}-{{e^ {- 1 }\,\left(4\,\sin \left({{1}\over{2}}\right) +2\,\cos \left({{1}\over{2}}\right)\right)}\over{5}}$$

       -1                        -1
2   4*e  *sin(1/2)   2*cos(1/2)*e  
- - -------------- - --------------
5         5                5

$$- \frac{4 \sin{\left(\frac{1}{2} \right)}}{5 e} - \frac{2 \cos{\left(\frac{1}{2} \right)}}{5 e} + \frac{2}{5}$$

Упростить

Numerical answer [src]

0.129765527639961

0.129765527639961

The graph

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

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Identical expressions

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Integral of (e^(-x))*(sin(x/2)) dx

Limits of integration:

The graph:

Piecewise:

The solution