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Similar expressions
2*x*exp(4*x+2)

Integral of 2xexp(4*x-2) dx

The solution

You have entered [src]

 1/2               
  /                
 |                 
 |       4*x - 2   
 |  2*x*e        dx
 |                 
/                  
0

$$\int\limits_{0}^{\frac{1}{2}} 2 x e^{4 x - 2}\, dx$$

Integral((2*x)*exp(4*x - 2), (x, 0, 1/2))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $2 x e^{4 x - 2} = \frac{2 x e^{4 x}}{e^{2}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 x e^{4 x}}{e^{2}}\, dx = \frac{2 \int x e^{4 x}\, dx}{e^{2}}$
  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 $\operatorname{dv}{\left(x \right)} = e^{4 x}$ .
    
    Then $\operatorname{du}{\left(x \right)} = 1$ .
    
    To find $v{\left(x \right)}$ :
    1. Let $u = 4 x$ .
      
      Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
      
      $\int \frac{e^{u}}{4}\, du$
      
      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: $\frac{e^{u}}{4}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{4 x}}{4}$
    Now evaluate the sub-integral.
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{e^{4 x}}{4}\, dx = \frac{\int e^{4 x}\, dx}{4}$
    1. Let $u = 4 x$ .
      
      Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
      
      $\int \frac{e^{u}}{4}\, du$
      
      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: $\frac{e^{u}}{4}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{4 x}}{4}$
    So, the result is: $\frac{e^{4 x}}{16}$
  So, the result is: $\frac{2 \left(\frac{x e^{4 x}}{4} - \frac{e^{4 x}}{16}\right)}{e^{2}}$
Method #2
1. Rewrite the integrand:
  
  $2 x e^{4 x - 2} = \frac{2 x e^{4 x}}{e^{2}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 x e^{4 x}}{e^{2}}\, dx = \frac{2 \int x e^{4 x}\, dx}{e^{2}}$
  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 $\operatorname{dv}{\left(x \right)} = e^{4 x}$ .
    
    Then $\operatorname{du}{\left(x \right)} = 1$ .
    
    To find $v{\left(x \right)}$ :
    1. Let $u = 4 x$ .
      
      Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
      
      $\int \frac{e^{u}}{4}\, du$
      
      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: $\frac{e^{u}}{4}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{4 x}}{4}$
    Now evaluate the sub-integral.
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{e^{4 x}}{4}\, dx = \frac{\int e^{4 x}\, dx}{4}$
    1. Let $u = 4 x$ .
      
      Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
      
      $\int \frac{e^{u}}{4}\, du$
      
      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: $\frac{e^{u}}{4}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{4 x}}{4}$
    So, the result is: $\frac{e^{4 x}}{16}$
  So, the result is: $\frac{2 \left(\frac{x e^{4 x}}{4} - \frac{e^{4 x}}{16}\right)}{e^{2}}$
Now simplify:

$\frac{\left(4 x - 1\right) e^{4 x - 2}}{8}$
Add the constant of integration:

$\frac{\left(4 x - 1\right) e^{4 x - 2}}{8}+ \mathrm{constant}$

The answer is:

$\frac{\left(4 x - 1\right) e^{4 x - 2}}{8}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                             
 |                         /   4*x      4*x\    
 |      4*x - 2            |  e      x*e   |  -2
 | 2*x*e        dx = C + 2*|- ---- + ------|*e  
 |                         \   16      4   /    
/

$$\int 2 x e^{4 x - 2}\, dx = C + \frac{2 \left(\frac{x e^{4 x}}{4} - \frac{e^{4 x}}{16}\right)}{e^{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

     -2
1   e  
- + ---
8    8

$$\frac{1}{8 e^{2}} + \frac{1}{8}$$

     -2
1   e  
- + ---
8    8

$$\frac{1}{8 e^{2}} + \frac{1}{8}$$

1/8 + exp(-2)/8

Numerical answer [src]

0.141916910404577

0.141916910404577

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of 2xexp(4*x-2) 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 2*x*exp(4*x-2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Integral of 2xexp(4*x-2) dx