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Integral of exp(-x/3)cos(x) dx

The solution

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 oo               
  /               
 |                
 |   -x           
 |   ---          
 |    3           
 |  e   *cos(x) dx
 |                
/                 
0

$$\int\limits_{0}^{\infty} e^{\frac{\left(-1\right) x}{3}} \cos{\left(x \right)}\, dx$$

Simplify

Detail solution

Use integration by parts, noting that the integrand eventually repeats itself.
1. For the integrand $e^{\frac{\left(-1\right) x}{3}} \cos{\left(x \right)}$ :
  
  Let $u{\left(x \right)} = \cos{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{\frac{\left(-1\right) x}{3}}$ .
  
  Then $\int e^{\frac{\left(-1\right) x}{3}} \cos{\left(x \right)}\, dx = - \int 3 e^{- \frac{x}{3}} \sin{\left(x \right)}\, dx - 3 e^{- \frac{x}{3}} \cos{\left(x \right)}$ .
2. For the integrand $3 e^{- \frac{x}{3}} \sin{\left(x \right)}$ :
  
  Let $u{\left(x \right)} = 3 \sin{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{- \frac{x}{3}}$ .
  
  Then $\int e^{\frac{\left(-1\right) x}{3}} \cos{\left(x \right)}\, dx = \int \left(- 9 e^{- \frac{x}{3}} \cos{\left(x \right)}\right)\, dx + 9 e^{- \frac{x}{3}} \sin{\left(x \right)} - 3 e^{- \frac{x}{3}} \cos{\left(x \right)}$ .
3. Notice that the integrand has repeated itself, so move it to one side:
  
  $10 \int e^{\frac{\left(-1\right) x}{3}} \cos{\left(x \right)}\, dx = 9 e^{- \frac{x}{3}} \sin{\left(x \right)} - 3 e^{- \frac{x}{3}} \cos{\left(x \right)}$
  
  Therefore,
  
  $\int e^{\frac{\left(-1\right) x}{3}} \cos{\left(x \right)}\, dx = \frac{9 e^{- \frac{x}{3}} \sin{\left(x \right)}}{10} - \frac{3 e^{- \frac{x}{3}} \cos{\left(x \right)}}{10}$
Now simplify:

$\frac{3 \cdot \left(3 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{- \frac{x}{3}}}{10}$
Add the constant of integration:

$\frac{3 \cdot \left(3 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{- \frac{x}{3}}}{10}+ \mathrm{constant}$

The answer is:

$\frac{3 \cdot \left(3 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{- \frac{x}{3}}}{10}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                  
 |                                -x       -x        
 |  -x                            ---      ---       
 |  ---                            3        3        
 |   3                  3*cos(x)*e      9*e   *sin(x)
 | e   *cos(x) dx = C - ------------- + -------------
 |                            10              10     
/

$${{9\,e^ {- {{x}\over{3}} }\,\left(\sin x-{{\cos x}\over{3}}\right) }\over{10}}$$

Simplify

The answer [src]

3/10

$$\frac{3}{10}$$

3/10

$$\frac{3}{10}$$

Simplify

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

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Integral of exp(-x/3)cos(x) dx

Limits of integration:

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Piecewise:

The solution