Integral of xexp(2x) dx

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

Integral(x*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 $u{\left(x \right)} = x$ 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:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

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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]

2.09726402473266

2.09726402473266

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Integral of xexp(2x) dx

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The solution