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Identical expressions
x*exp(three *x- five)
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x*exp(3*x-5)dx
Similar expressions
x*exp(3*x+5)

Integral of xexp(3x-5) dx

The solution

You have entered [src]

  1              
  /              
 |               
 |     3*x - 5   
 |  x*e        dx
 |               
/                
0

\int\limits_{0}^{1} x e^{3 x - 5}\, dx

Integral(x*exp(3*x - 5), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $x e^{3 x - 5} = \frac{x e^{3 x}}{e^{5}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x e^{3 x}}{e^{5}}\, dx = \frac{\int x e^{3 x}\, dx}{e^{5}}$
  1. 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 $\operatorname{dv}{\left(x \right)} = e^{3 x}$ .
    
    Then $\operatorname{du}{\left(x \right)} = 1$ .
    
    To find $v{\left(x \right)}$ :
    1. Let $u = 3 x$ .
      
      Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
      
      $\int \frac{e^{u}}{3}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{3 x}}{3}$
    Now evaluate the sub-integral.
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{e^{3 x}}{3}\, dx = \frac{\int e^{3 x}\, dx}{3}$
    1. Let $u = 3 x$ .
      
      Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
      
      $\int \frac{e^{u}}{3}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{3 x}}{3}$
    So, the result is: $\frac{e^{3 x}}{9}$
  So, the result is: $\frac{\frac{x e^{3 x}}{3} - \frac{e^{3 x}}{9}}{e^{5}}$
Method #2
1. Rewrite the integrand:
  
  $x e^{3 x - 5} = \frac{x e^{3 x}}{e^{5}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x e^{3 x}}{e^{5}}\, dx = \frac{\int x e^{3 x}\, dx}{e^{5}}$
  1. 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 $\operatorname{dv}{\left(x \right)} = e^{3 x}$ .
    
    Then $\operatorname{du}{\left(x \right)} = 1$ .
    
    To find $v{\left(x \right)}$ :
    1. Let $u = 3 x$ .
      
      Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
      
      $\int \frac{e^{u}}{3}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{3 x}}{3}$
    Now evaluate the sub-integral.
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{e^{3 x}}{3}\, dx = \frac{\int e^{3 x}\, dx}{3}$
    1. Let $u = 3 x$ .
      
      Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
      
      $\int \frac{e^{u}}{3}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{3 x}}{3}$
    So, the result is: $\frac{e^{3 x}}{9}$
  So, the result is: $\frac{\frac{x e^{3 x}}{3} - \frac{e^{3 x}}{9}}{e^{5}}$
Now simplify:

$\frac{\left(3 x - 1\right) e^{3 x - 5}}{9}$
Add the constant of integration:

$\frac{\left(3 x - 1\right) e^{3 x - 5}}{9}+ \mathrm{constant}$

The answer is:

\frac{\left(3 x - 1\right) e^{3 x - 5}}{9}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                                         
 |                     /   3*x      3*x\    
 |    3*x - 5          |  e      x*e   |  -5
 | x*e        dx = C + |- ---- + ------|*e  
 |                     \   9       3   /    
/

\int x e^{3 x - 5}\, dx = C + \frac{\frac{x e^{3 x}}{3} - \frac{e^{3 x}}{9}}{e^{5}}

Simplify

The graph

Plot the graph f(x)

The answer [src]

 -5      -2
e     2*e  
--- + -----
 9      9

\frac{1}{9 e^{5}} + \frac{2}{9 e^{2}}

 -5      -2
e     2*e  
--- + -----
 9      9

\frac{1}{9 e^{5}} + \frac{2}{9 e^{2}}

exp(-5)/9 + 2*exp(-2)/9

Numerical answer [src]

0.0308231681635901

0.0308231681635901

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of xexp(3x-5) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of x*exp(3*x-5) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Integral of xexp(3x-5) dx