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Identical expressions
(3x+ four)*e^(3x)
(3x plus 4) multiply by e to the power of (3x)
(3x plus four) multiply by e to the power of (3x)
(3x+4)*e(3x)
3x+4*e3x
(3x+4)e^(3x)
(3x+4)e(3x)
3x+4e3x
3x+4e^3x
(3x+4)*e^(3x)dx
Similar expressions
(3x-4)*e^(3x)

Solving integrals/ (3x+4)*e^(3x)

Integral of (3x+4)*e^(3x) dx

The solution

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  1                  
  /                  
 |                   
 |             3*x   
 |  (3*x + 4)*E    dx
 |                   
/                    
0

\int\limits_{0}^{1} e^{3 x} \left(3 x + 4\right)\, dx

Integral((3*x + 4)*E^(3*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $du$ :
  
  $\int \left(\frac{u e^{u}}{3} + \frac{4 e^{u}}{3}\right)\, du$
  1. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u e^{u}}{3}\, du = \frac{\int u e^{u}\, du}{3}$
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\operatorname{du}{\left(u \right)} = 1$ .
      
      To find $v{\left(u \right)}$ :
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      Now evaluate the sub-integral.
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{u e^{u}}{3} - \frac{e^{u}}{3}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{4 e^{u}}{3}\, du = \frac{4 \int e^{u}\, du}{3}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{4 e^{u}}{3}$
    The result is: $\frac{u e^{u}}{3} + e^{u}$
  Now substitute $u$ back in:
  
  $x e^{3 x} + e^{3 x}$
Method #2
1. Rewrite the integrand:
  
  $e^{3 x} \left(3 x + 4\right) = 3 x e^{3 x} + 4 e^{3 x}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 x e^{3 x}\, dx = 3 \int x e^{3 x}\, dx$
    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 $v{\left(x \right)}$ :
      
      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:
      
      $\int \frac{e^{3 x}}{3}\, dx = \frac{\int e^{3 x}\, dx}{3}$
      
      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: $x e^{3 x} - \frac{e^{3 x}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 4 e^{3 x}\, dx = 4 \int e^{3 x}\, dx$
    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: $\frac{4 e^{3 x}}{3}$
  The result is: $x e^{3 x} + e^{3 x}$
Method #3
1. Rewrite the integrand:
  
  $e^{3 x} \left(3 x + 4\right) = 3 x e^{3 x} + 4 e^{3 x}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 x e^{3 x}\, dx = 3 \int x e^{3 x}\, dx$
    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 $v{\left(x \right)}$ :
      
      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:
      
      $\int \frac{e^{3 x}}{3}\, dx = \frac{\int e^{3 x}\, dx}{3}$
      
      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: $x e^{3 x} - \frac{e^{3 x}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 4 e^{3 x}\, dx = 4 \int e^{3 x}\, dx$
    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: $\frac{4 e^{3 x}}{3}$
  The result is: $x e^{3 x} + e^{3 x}$
Now simplify:

$\left(x + 1\right) e^{3 x}$
Add the constant of integration:

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

The answer is:

\left(x + 1\right) e^{3 x}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                                     
 |                                      
 |            3*x             3*x    3*x
 | (3*x + 4)*E    dx = C + x*e    + e   
 |                                      
/

\int e^{3 x} \left(3 x + 4\right)\, dx = C + x e^{3 x} + e^{3 x}

Simplify

The graph

Plot the graph f(x)

The answer [src]

        3
-1 + 2*e

-1 + 2 e^{3}

        3
-1 + 2*e

-1 + 2 e^{3}

-1 + 2*exp(3)

Numerical answer [src]

39.1710738463753

39.1710738463753

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 (3x+4)*e^(3x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3