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Identical expressions
e^(-x)*sen(x)
e to the power of ( minus x) multiply by sen(x)
e(-x)*sen(x)
e-x*senx
e^(-x)sen(x)
e(-x)sen(x)
e-xsenx
e^-xsenx
e^(-x)*sen(x)dx
Similar expressions
e^(x)*sen(x)

Integral of e^(-x)*sen(x) dx

The solution

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 oo              
  /              
 |               
 |   -x          
 |  e  *sin(x) dx
 |               
/                
-oo

$$\int\limits_{-\infty}^{\infty} e^{- x} \sin{\left(x \right)}\, dx$$

Integral(sin(x)/E^x, (x, -oo, oo))

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

<-oo, oo>

$$-{{e^{{\it oo}}\,\sin {\it oo}-e^{{\it oo}}\,\cos {\it oo}}\over{2 }}-{{e^ {- {\it oo} }\,\left(\sin {\it oo}+\cos {\it oo}\right) }\over{2}}$$

<-oo, oo>

$$\left\langle -\infty, \infty\right\rangle$$

Упростить

Numerical answer [src]

-2.30141637303258e+4333359448156331752

-2.30141637303258e+4333359448156331752

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

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Identical expressions

Similar expressions

Integral of e^(-x)*sen(x) dx

Limits of integration:

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Piecewise:

The solution