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Solving integrals/ x^3exp(-2x^2)

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

Limits of integration:

from to
v

The graph:

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Piecewise:

The solution

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∫01x3e−2x2 dx\int\limits_{0}^{1} x^{3} e^{- 2 x^{2}}\, dx 
Integral(x^3*exp(-2*x^2), (x, 0, 1))
Detail solution

  1. Use integration by parts:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=x3u{\left(x \right)} = x^{3} and let dv⁡(x)=e−2x2\operatorname{dv}{\left(x \right)} = e^{- 2 x^{2}}.

    Then du⁡(x)=3x2\operatorname{du}{\left(x \right)} = 3 x^{2}.

    To find v(x)v{\left(x \right)}:

      ErfRule(a=-2, b=0, c=0, context=exp(-2*x**2), symbol=x)

    Now evaluate the sub-integral.

  2. The integral of a constant times a function is the constant times the integral of the function:

    ∫32πx2erf⁡(2x)4 dx=32π∫x2erf⁡(2x) dx4\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

      x3erf⁡(2x)3+2x2e−2x26π+2e−2x212π\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: 32π(x3erf⁡(2x)3+2x2e−2x26π+2e−2x212π)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}

  3. Now simplify:

    −(2x2+1)e−2x28- \frac{\left(2 x^{2} + 1\right) e^{- 2 x^{2}}}{8}

  4. Add the constant of integration:

    −(2x2+1)e−2x28+constant- \frac{\left(2 x^{2} + 1\right) e^{- 2 x^{2}}}{8}+ \mathrm{constant}

The answer is:

−(2x2+1)e−2x28+constant- \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              
/
∫x3e−2x2 dx=C+2πx3erf⁡(2x)4−32π(x3erf⁡(2x)3+2x2e−2x26π+2e−2x212π)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
0.001.000.100.200.300.400.500.600.700.800.900.25-0.25
Plot the graph f(x)
The answer [src] 
       -2
1   3*e  
- - -----
8     8
18−38e2\frac{1}{8} - \frac{3}{8 e^{2}} 
=
= 
       -2
1   3*e  
- - -----
8     8
18−38e2\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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