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Integral of d{x}:
Integral of x^1
Integral of sin(x)/cos(x)
Integral of -2*x
Integral of e^(x+2)
Derivative of:
(x^2)*exp(2x)
Identical expressions
(x^ two)*exp(2x)
(x squared ) multiply by exponent of (2x)
(x to the power of two) multiply by exponent of (2x)
(x2)*exp(2x)
x2*exp2x
(x²)*exp(2x)
(x to the power of 2)*exp(2x)
(x^2)exp(2x)
(x2)exp(2x)
x2exp2x
x^2exp2x
(x^2)*exp(2x)dx

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

The solution

You have entered [src]

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

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

Integral(x^2*exp(2*x), (x, 0, 1))

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. Let $u = 2 x$ .
  
  Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{e^{u}}{2}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    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.
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 $\frac{du}{2}$ :
  
  $\int \frac{e^{u}}{2}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    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 $\frac{du}{2}$ :
  
  $\int \frac{e^{u}}{2}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    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   
/

$$\int x^{2} e^{2 x}\, dx = C + \frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{2 x}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

-1/4 + exp(2)/4

Numerical answer [src]

1.59726402473266

1.59726402473266

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)*exp(2x) dx

Limits of integration:

The graph:

Piecewise:

The solution