Question

Evaluate `int_(-oo)^(oo) \ x^2e^(-4x) \ dx` ?

Answer 1

The integral is divergent.

Explanation:

We seek:

`int_(-oo)^(oo) \ x^2e^(-4x) \ dx`

Using common sense analysis we can answer the question without actually performing the integral.

If we apply Integration By Parts twice we will get a solution of the definite integral of the form:

`int \ x^2e^(-4x) \ dx = (Ax^2+Bx+C)e^(-4x) + C`

Thuis we will find that:

`int_(-oo)^(oo) \ x^2e^(-4x) \ dx = lim_(N rarr oo)[(Ax^2+Bx+C)e^(-4x)]_(-N)^(N)`

And will will get convergence at the upper limit as `e^(-oo) rarr 0` but divergence at the lower limit as `e^(oo) rarr 0`,

Hence the integral is divergent.

Note:

Two applications of IBP yields:

`int \ x^2e^(-4x) \ dx = -((8x^2+4x+1)e^(-4x))/32 + C`