1 / | | 2 | 3 -2*x | x *e dx | / 0
Integral(x^3*exp(-2*x^2), (x, 0, 1))
Use integration by parts:
Let and let .
Then .
To find :
ErfRule(a=-2, b=0, c=0, context=exp(-2*x**2), symbol=x)
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:
Don't know the steps in finding this integral.
But the integral is
So, the result is:
Now simplify:
Add the constant of integration:
The answer is:
/ 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 /
-2 1 3*e - - ----- 8 8
=
-2 1 3*e - - ----- 8 8
1/8 - 3*exp(-2)/8
Use the examples entering the upper and lower limits of integration.