I Solving Integral for All n≥2 | Evans PDE's (Page 48)

[NanoMath]

In the book from Evans on PDE's (page 48) I came across this integral. Here r > 0 and \delta is an arbitrarily small number.
Could you give me some hint on how to solve this integral for all integers n\geq2, i.e why does it go to zero as t approaches zero from the right side.

 

[eys_physics]

Try to a change of variable of the form $$x=r/\sqrt{16t}$$.

 

[mathman]

Let K(t)=e^{\frac{-\delta^2}{16t}}. Put K as a multiplier outside the integral sign, and 1/K inside. The integral is bounded as t ->0+, while the coefficient -> 0. As for the integral itself, it looks like it will include the error function as a term.

 

[NanoMath]

Thank you for replies. Actually I only needed to check that the whole expression goes to zero and multiplying and dividing through by K(t) solves it.