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Solving integrals/ e^(-x)sinx

Integral of e^(-x)sinx dx

The solution

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  1              
  /              
 |               
 |   -x          
 |  E  *sin(x) dx
 |               
/                
0

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

Integral(E^(-x)*sin(x), (x, 0, 1))

Detail solution

Let $u = - x$ .

Then let $du = - dx$ and substitute $du$ :

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

$- \frac{e^{- x} \sin{\left(x \right)}}{2} - \frac{e^{- x} \cos{\left(x \right)}}{2}$
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     
/

$$\int e^{- x} \sin{\left(x \right)}\, dx = C - \frac{e^{- x} \sin{\left(x \right)}}{2} - \frac{e^{- x} \cos{\left(x \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

1/2 - cos(1)*exp(-1)/2 - exp(-1)*sin(1)/2

Numerical answer [src]

0.245837007000237

0.245837007000237

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)sinx dx

Limits of integration:

The graph:

Piecewise:

The solution