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Integral of d{x}:
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Integral of lnx^2
Identical expressions
x^(two)*e^(2x)
x to the power of (2) multiply by e to the power of (2x)
x to the power of (two) multiply by e to the power of (2x)
x(2)*e(2x)
x2*e2x
x^(2)e^(2x)
x(2)e(2x)
x2e2x
x^2e^2x
x^(2)*e^(2x)dx

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

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

The solution

You have entered [src]

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

\int\limits_{0}^{1} x^{2} e^{2 x}\, dx

Simplify

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = x^{2}$ and let $\operatorname{dv}{\left(x \right)} = e^{2 x}$ .

Then $\operatorname{du}{\left(x \right)} = 2 x$ .

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

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

To find $v{\left(x \right)}$ :
1. Let $u = 2 x$ .
  
  Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{e^{u}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{e^{u}}{2}\, du = \frac{\int e^{u}\, du}{2}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $\frac{e^{u}}{2}$
  Now substitute $u$ back in:
  
  $\frac{e^{2 x}}{2}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{e^{2 x}}{2}\, dx = \frac{\int e^{2 x}\, dx}{2}$
1. Let $u = 2 x$ .
  
  Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{e^{u}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{e^{u}}{2}\, du = \frac{\int e^{u}\, du}{2}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $\frac{e^{u}}{2}$
  Now substitute $u$ back in:
  
  $\frac{e^{2 x}}{2}$
So, the result is: $\frac{e^{2 x}}{4}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

{{\left(2\,x^2-2\,x+1\right)\,e^{2\,x}}\over{4}}

Simplify

The graph

Plot the graph f(x)

The answer [src]

{{e^2}\over{4}}-{{1}\over{4}}

- \frac{1}{4} + \frac{e^{2}}{4}

Simplify

Numerical answer [src]

1.59726402473266

1.59726402473266

The graph

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

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How to use it?

Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2