Evaluate $I(a) = \int_{0}^{\infty}\frac{e^{-ax^2}-e^{-x^2}}{x}dx$

I suspect that this problem involves Feynman's trick for integration but am unsure.

Evaluate $I(a) = \int_{0}^{\infty}\frac{e^{-ax^2}-e^{-x^2}}{x}dx$

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Kav10
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• @Kav10: You solution was incomplete and did not answer the question. I added an answer to the end of your solution. Please make sure that you can submit a complete solution before accepting similar questions in the future.

• Thank you!

• @Walter, your idea is good but I think you made a mistake in the differentiation, and that makes everything after that wrong! I have uploaded the correct differentiation and the rest of the solution.

• You are right, thank you for pointing that out. I revised my solution. You may want to remove the first solution you submitted as it may be confusing to Curious Math Geek.