Efficient Integration of Exponential Functions with Infinite Limits

[ronaldoshaky]

Homework Statement

how would you integrate, int (-ik - 2ax)*exp(-2ax^2) dx with limits infinity - infinity

Homework Equations

i think i can use the result int exp(-x^2) dx = sqrt (pi). But I am stumped.

The Attempt at a Solution

i thought about multiplying it out

int -ik*exp(-2ax^2) - 2ax*exp(-2ax^2) dx with limits infinity and - infinity

but it looks more complex now. would you use integration by substitution

Thank you

 

[tiny-tim]

Welcome to PF!

Hi ronaldoshaky! Welcome to PF!

(have an integral: ∫ and an infinity: ∞ and a pi: π and try using the X² tag just above the Reply box )

Just do the two parts separately …

∫_-∞^∞ ik e^-2ax2 dx you know how to do;

and ∫ 2ax e^-2ax2 dx you can do by substitution.

 

[ronaldoshaky]

Thank you!

 

[ronaldoshaky]

Hi tiny-tim

Can I evaluate this integral ∫-∞∞ (ik-2ax) exp(-2ax2) dx without multiplying it out,
using integration by parts because it is a product?

I get


∫-∞∞ (ik-2ax) exp(-2ax2) dx = f(x)g(x) - ∫ f '(x)g(x) = - ik (sqrt (pi/2a)) + ikx (sqrt (pi/2a))

Is that any way correct?

 

[tiny-tim]

No.