Proving two integrals are equal using substitution

[cosmic123]

I have two integrals, and I need to prove the first one is equal to the second using the substitution x=sqrt(u). Here is the answer I got, but am unsure of my algebra to transform the function to sqrt(1+1/4u).

 

[tkhunny]

We're going to struggle, here, with "is equal to".

1) We should notice that the Domains are different. The 1/(4x) integrand has a big hole in it. Maybe we mean "is equal to" only for Positive Real Numbers?

2) Really, our only hope of claiming these two integrals are equal is if they differ by only a constant. That doesn't quite sound "equal", does it? Ponder why an integral which evaluates to $\displaystyle \cos^{2}(x) + C$ is EXACTLY the same as an integral that evaluates to $\displaystyle \sin^{2}(x) + D$.

3) You started your problem with $\displaystyle x = \sqrt{u}$. You ended your demonstration with $\displaystyle x = u$. That's no good.

Good work. Give it some more thought. In particular, think about whether it CAN be true before spending a lot of time trying to prove it.

 

[cosmic123]

We're going to struggle, here, with "is equal to".

1) We should notice that the Domains are different. The 1/(4x) integrand has a big hole in it. Maybe we mean "is equal to" only for Positive Real Numbers?

2) Really, our only hope of claiming these two integrals are equal is if they differ by only a constant. That doesn't quite sound "equal", does it? Ponder why an integral which evaluates to $\displaystyle \cos^{2}(x) + C$ is EXACTLY the same as an integral that evaluates to $\displaystyle \sin^{2}(x) + D$.

3) You started your problem with $\displaystyle x = \sqrt{u}$. You ended your demonstration with $\displaystyle x = u$. That's no good.

Good work. Give it some more thought. In particular, think about whether it CAN be true before spending a lot of time trying to prove it.

I'm sorry, the exact wording for the question was "Show that one integral can be transformed into the other."

 

[tkhunny]

Well, then I guess you're done. Just heed comment #3.