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Identical expressions
x^3exp(- two x^2)
x cubed exponent of ( minus 2x squared )
x cubed exponent of ( minus two x squared )
x3exp(-2x2)
x3exp-2x2
x³exp(-2x²)
x to the power of 3exp(-2x to the power of 2)
x^3exp-2x^2
x^3exp(-2x^2)dx
Similar expressions
x^3exp(2x^2)

Integral of x^3exp(-2x^2) dx

The solution

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  1             
  /             
 |              
 |          2   
 |   3  -2*x    
 |  x *e      dx
 |              
/               
0

$$\int\limits_{0}^{1} x^{3} e^{- 2 x^{2}}\, dx$$

Integral(x^3*exp(-2*x^2), (x, 0, 1))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = x^{3}$ and let $\operatorname{dv}{\left(x \right)} = e^{- 2 x^{2}}$ .

Then $\operatorname{du}{\left(x \right)} = 3 x^{2}$ .

To find $v{\left(x \right)}$ :
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{3 \sqrt{2} \sqrt{\pi} x^{2} \operatorname{erf}{\left(\sqrt{2} x \right)}}{4}\, dx = \frac{3 \sqrt{2} \sqrt{\pi} \int x^{2} \operatorname{erf}{\left(\sqrt{2} x \right)}\, dx}{4}$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{x^{3} \operatorname{erf}{\left(\sqrt{2} x \right)}}{3} + \frac{\sqrt{2} x^{2} e^{- 2 x^{2}}}{6 \sqrt{\pi}} + \frac{\sqrt{2} e^{- 2 x^{2}}}{12 \sqrt{\pi}}$
So, the result is: $\frac{3 \sqrt{2} \sqrt{\pi} \left(\frac{x^{3} \operatorname{erf}{\left(\sqrt{2} x \right)}}{3} + \frac{\sqrt{2} x^{2} e^{- 2 x^{2}}}{6 \sqrt{\pi}} + \frac{\sqrt{2} e^{- 2 x^{2}}}{12 \sqrt{\pi}}\right)}{4}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

                                     /                             2                 2\                               
                                     | 3    /    ___\     ___  -2*x      ___  2  -2*x |                               
  /                       ___   ____ |x *erf\x*\/ 2 /   \/ 2 *e        \/ 2 *x *e     |                               
 |                    3*\/ 2 *\/ pi *|--------------- + ------------ + ---------------|                               
 |         2                         |       3                ____             ____   |     ___   ____  3    /    ___\
 |  3  -2*x                          \                   12*\/ pi          6*\/ pi    /   \/ 2 *\/ pi *x *erf\x*\/ 2 /
 | x *e      dx = C - ----------------------------------------------------------------- + ----------------------------
 |                                                    4                                                4              
/

$$\int x^{3} e^{- 2 x^{2}}\, dx = C + \frac{\sqrt{2} \sqrt{\pi} x^{3} \operatorname{erf}{\left(\sqrt{2} x \right)}}{4} - \frac{3 \sqrt{2} \sqrt{\pi} \left(\frac{x^{3} \operatorname{erf}{\left(\sqrt{2} x \right)}}{3} + \frac{\sqrt{2} x^{2} e^{- 2 x^{2}}}{6 \sqrt{\pi}} + \frac{\sqrt{2} e^{- 2 x^{2}}}{12 \sqrt{\pi}}\right)}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       -2
1   3*e  
- - -----
8     8

$$\frac{1}{8} - \frac{3}{8 e^{2}}$$

       -2
1   3*e  
- - -----
8     8

$$\frac{1}{8} - \frac{3}{8 e^{2}}$$

1/8 - 3*exp(-2)/8

Numerical answer [src]

0.0742492687862702

0.0742492687862702

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

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Integral of x^3exp(-2x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution