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Identical expressions
(x+ two)*e^(three *x)
(x plus 2) multiply by e to the power of (3 multiply by x)
(x plus two) multiply by e to the power of (three multiply by x)
(x+2)*e(3*x)
x+2*e3*x
(x+2)e^(3x)
(x+2)e(3x)
x+2e3x
x+2e^3x
(x+2)*e^(3*x)dx
Similar expressions
(x-2)*e^(3*x)

Solving integrals/ (x+2)*e^(3*x)

Integral of (x+2)e^(3x) dx

The solution

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  1                
  /                
 |                 
 |           3*x   
 |  (x + 2)*E    dx
 |                 
/                  
0

$$\int\limits_{0}^{1} e^{3 x} \left(x + 2\right)\, dx$$

Integral((x + 2)*E^(3*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $e^{3 x} \left(x + 2\right) = x e^{3 x} + 2 e^{3 x}$
2. Integrate term-by-term:
  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: $\frac{e^{3 x}}{9}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 e^{3 x}\, dx = 2 \int e^{3 x}\, dx$
    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: $\frac{2 e^{3 x}}{3}$
  The result is: $\frac{x e^{3 x}}{3} + \frac{5 e^{3 x}}{9}$
Method #2
1. Rewrite the integrand:
  
  $e^{3 x} \left(x + 2\right) = x e^{3 x} + 2 e^{3 x}$
2. Integrate term-by-term:
  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: $\frac{e^{3 x}}{9}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 e^{3 x}\, dx = 2 \int e^{3 x}\, dx$
    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: $\frac{2 e^{3 x}}{3}$
  The result is: $\frac{x e^{3 x}}{3} + \frac{5 e^{3 x}}{9}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                     
 |                          3*x      3*x
 |          3*x          5*e      x*e   
 | (x + 2)*E    dx = C + ------ + ------
 |                         9        3   
/

$$\int e^{3 x} \left(x + 2\right)\, dx = C + \frac{x e^{3 x}}{3} + \frac{5 e^{3 x}}{9}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

         3
  5   8*e 
- - + ----
  9    9

$$- \frac{5}{9} + \frac{8 e^{3}}{9}$$

         3
  5   8*e 
- - + ----
  9    9

$$- \frac{5}{9} + \frac{8 e^{3}}{9}$$

-5/9 + 8*exp(3)/9

Numerical answer [src]

17.2982550428335

17.2982550428335

The graph

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

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How to use it?

Identical expressions

Similar expressions

Integral of (x+2)e^(3x) 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+2)*e^(3*x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Integral of (x+2)e^(3x) dx