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xe^(-3x)cos(5x)
xe to the power of ( minus 3x) co sinus of e of (5x)
xe(-3x)cos(5x)
xe-3xcos5x
xe^-3xcos5x
xe^(-3x)cos(5x)dx
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xe^(3x)cos(5x)

Integral of xe^(-3x)cos(5x) dx

The solution

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 oo                    
  /                    
 |                     
 |     -3*x            
 |  x*E    *cos(5*x) dx
 |                     
/                      
0

$$\int\limits_{0}^{\infty} e^{- 3 x} x \cos{\left(5 x \right)}\, dx$$

Integral((x*E^(-3*x))*cos(5*x), (x, 0, oo))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = e^{- 3 x} \cos{\left(5 x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 1$ .

To find $v{\left(x \right)}$ :
1. Use integration by parts, noting that the integrand eventually repeats itself.
  1. For the integrand $e^{- 3 x} \cos{\left(5 x \right)}$ :
    
    Let $u{\left(x \right)} = \cos{\left(5 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{- 3 x}$ .
    
    Then $\int e^{- 3 x} \cos{\left(5 x \right)}\, dx = - \int \frac{5 e^{- 3 x} \sin{\left(5 x \right)}}{3}\, dx - \frac{e^{- 3 x} \cos{\left(5 x \right)}}{3}$ .
  2. For the integrand $\frac{5 e^{- 3 x} \sin{\left(5 x \right)}}{3}$ :
    
    Let $u{\left(x \right)} = \frac{5 \sin{\left(5 x \right)}}{3}$ and let $\operatorname{dv}{\left(x \right)} = e^{- 3 x}$ .
    
    Then $\int e^{- 3 x} \cos{\left(5 x \right)}\, dx = \int \left(- \frac{25 e^{- 3 x} \cos{\left(5 x \right)}}{9}\right)\, dx + \frac{5 e^{- 3 x} \sin{\left(5 x \right)}}{9} - \frac{e^{- 3 x} \cos{\left(5 x \right)}}{3}$ .
  3. Notice that the integrand has repeated itself, so move it to one side:
    
    $\frac{34 \int e^{- 3 x} \cos{\left(5 x \right)}\, dx}{9} = \frac{5 e^{- 3 x} \sin{\left(5 x \right)}}{9} - \frac{e^{- 3 x} \cos{\left(5 x \right)}}{3}$
    
    Therefore,
    
    $\int e^{- 3 x} \cos{\left(5 x \right)}\, dx = \frac{5 e^{- 3 x} \sin{\left(5 x \right)}}{34} - \frac{3 e^{- 3 x} \cos{\left(5 x \right)}}{34}$
Now evaluate the sub-integral.
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{5 e^{- 3 x} \sin{\left(5 x \right)}}{34}\, dx = \frac{5 \int e^{- 3 x} \sin{\left(5 x \right)}\, dx}{34}$
  1. Use integration by parts, noting that the integrand eventually repeats itself.
    1. For the integrand $e^{- 3 x} \sin{\left(5 x \right)}$ :
      
      Let $u{\left(x \right)} = \sin{\left(5 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{- 3 x}$ .
      
      Then $\int e^{- 3 x} \sin{\left(5 x \right)}\, dx = - \int \left(- \frac{5 e^{- 3 x} \cos{\left(5 x \right)}}{3}\right)\, dx - \frac{e^{- 3 x} \sin{\left(5 x \right)}}{3}$ .
    2. For the integrand $- \frac{5 e^{- 3 x} \cos{\left(5 x \right)}}{3}$ :
      
      Let $u{\left(x \right)} = - \frac{5 \cos{\left(5 x \right)}}{3}$ and let $\operatorname{dv}{\left(x \right)} = e^{- 3 x}$ .
      
      Then $\int e^{- 3 x} \sin{\left(5 x \right)}\, dx = \int \left(- \frac{25 e^{- 3 x} \sin{\left(5 x \right)}}{9}\right)\, dx - \frac{e^{- 3 x} \sin{\left(5 x \right)}}{3} - \frac{5 e^{- 3 x} \cos{\left(5 x \right)}}{9}$ .
    3. Notice that the integrand has repeated itself, so move it to one side:
      
      $\frac{34 \int e^{- 3 x} \sin{\left(5 x \right)}\, dx}{9} = - \frac{e^{- 3 x} \sin{\left(5 x \right)}}{3} - \frac{5 e^{- 3 x} \cos{\left(5 x \right)}}{9}$
      
      Therefore,
      
      $\int e^{- 3 x} \sin{\left(5 x \right)}\, dx = - \frac{3 e^{- 3 x} \sin{\left(5 x \right)}}{34} - \frac{5 e^{- 3 x} \cos{\left(5 x \right)}}{34}$
  So, the result is: $- \frac{15 e^{- 3 x} \sin{\left(5 x \right)}}{1156} - \frac{25 e^{- 3 x} \cos{\left(5 x \right)}}{1156}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{3 e^{- 3 x} \cos{\left(5 x \right)}}{34}\right)\, dx = - \frac{3 \int e^{- 3 x} \cos{\left(5 x \right)}\, dx}{34}$
  1. Use integration by parts, noting that the integrand eventually repeats itself.
    1. For the integrand $e^{- 3 x} \cos{\left(5 x \right)}$ :
      
      Let $u{\left(x \right)} = \cos{\left(5 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{- 3 x}$ .
      
      Then $\int e^{- 3 x} \cos{\left(5 x \right)}\, dx = - \int \frac{5 e^{- 3 x} \sin{\left(5 x \right)}}{3}\, dx - \frac{e^{- 3 x} \cos{\left(5 x \right)}}{3}$ .
    2. For the integrand $\frac{5 e^{- 3 x} \sin{\left(5 x \right)}}{3}$ :
      
      Let $u{\left(x \right)} = \frac{5 \sin{\left(5 x \right)}}{3}$ and let $\operatorname{dv}{\left(x \right)} = e^{- 3 x}$ .
      
      Then $\int e^{- 3 x} \cos{\left(5 x \right)}\, dx = \int \left(- \frac{25 e^{- 3 x} \cos{\left(5 x \right)}}{9}\right)\, dx + \frac{5 e^{- 3 x} \sin{\left(5 x \right)}}{9} - \frac{e^{- 3 x} \cos{\left(5 x \right)}}{3}$ .
    3. Notice that the integrand has repeated itself, so move it to one side:
      
      $\frac{34 \int e^{- 3 x} \cos{\left(5 x \right)}\, dx}{9} = \frac{5 e^{- 3 x} \sin{\left(5 x \right)}}{9} - \frac{e^{- 3 x} \cos{\left(5 x \right)}}{3}$
      
      Therefore,
      
      $\int e^{- 3 x} \cos{\left(5 x \right)}\, dx = \frac{5 e^{- 3 x} \sin{\left(5 x \right)}}{34} - \frac{3 e^{- 3 x} \cos{\left(5 x \right)}}{34}$
  So, the result is: $- \frac{15 e^{- 3 x} \sin{\left(5 x \right)}}{1156} + \frac{9 e^{- 3 x} \cos{\left(5 x \right)}}{1156}$
The result is: $- \frac{15 e^{- 3 x} \sin{\left(5 x \right)}}{578} - \frac{4 e^{- 3 x} \cos{\left(5 x \right)}}{289}$
Now simplify:

$\frac{\left(85 x \sin{\left(5 x \right)} - 51 x \cos{\left(5 x \right)} + 15 \sin{\left(5 x \right)} + 8 \cos{\left(5 x \right)}\right) e^{- 3 x}}{578}$
Add the constant of integration:

$\frac{\left(85 x \sin{\left(5 x \right)} - 51 x \cos{\left(5 x \right)} + 15 \sin{\left(5 x \right)} + 8 \cos{\left(5 x \right)}\right) e^{- 3 x}}{578}+ \mathrm{constant}$

The answer is:

$\frac{\left(85 x \sin{\left(5 x \right)} - 51 x \cos{\left(5 x \right)} + 15 \sin{\left(5 x \right)} + 8 \cos{\left(5 x \right)}\right) e^{- 3 x}}{578}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                          
 |                             /              -3*x      -3*x         \               -3*x       -3*x         
 |    -3*x                     |  3*cos(5*x)*e       5*e    *sin(5*x)|   4*cos(5*x)*e       15*e    *sin(5*x)
 | x*E    *cos(5*x) dx = C + x*|- ---------------- + ----------------| + ---------------- + -----------------
 |                             \         34                 34       /         289                 578       
/

$$\int e^{- 3 x} x \cos{\left(5 x \right)}\, dx = C + x \left(\frac{5 e^{- 3 x} \sin{\left(5 x \right)}}{34} - \frac{3 e^{- 3 x} \cos{\left(5 x \right)}}{34}\right) + \frac{15 e^{- 3 x} \sin{\left(5 x \right)}}{578} + \frac{4 e^{- 3 x} \cos{\left(5 x \right)}}{289}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-4/289

$$- \frac{4}{289}$$

-4/289

$$- \frac{4}{289}$$

-4/289

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

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Identical expressions

Similar expressions

Integral of xe^(-3x)cos(5x) dx

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The solution