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I'm having a tough time with an integral, and I'm not sure if I'm just not thinking about it right. To get full context, here is the first part of the question (and my solution to that first part, as it seems like it will be used in the part I'm having trouble with): enter image description here

My solution to part (a):

$$a_{m,Q}=\left(\frac{\Omega_{m,0}}{1-\Omega_{m,0}}\right)^{2/3}$$

Now the question I'm getting stuck on: enter image description here

This is what I've done so far, but when I get to this integral, I'm not sure how to get it to look like the solution the question says it should look like from eyeballing it.

enter image description here

Can I get any nudges in the right direction of where to go next? For instance, even if I just throw my hands up and put this into WolframAlpha, the solution doesn't look anything like what it "should" per the question. If I try to do it by hand, I find my solution drifts towards looking like WolframAlpha does.

This is what WolframAlpha does just for comparison (it will not do the definite integral from 0 to a, it runs out of compute time. But, even eyeballing this and imagining doing the fundamental theorem of calculus and doing the subtraction, it wouldn't look like the provided solution).

WolframAlpha's solution:

$$\int\frac{\sqrt{a}}{\sqrt{a^{3/2}(1-b)+b}}da=-\frac{4\sqrt{b-a^{3/2}(b-1)}}{3(b-1)}+\text{constant}$$

Thank you for any help :)

Edit: Also I should point out that when doing the fundamental theorem of calculus by hand, I can see hints of stuff that looks like the provided solution. For instance the numerator on one side of the minus sign will have the 4sqrt(omega_m)/3(1 - omega_m). But the rest will look nothing like the solution we're supposed to get. So I feel like I'm on the verge of some understanding, but just... missing it.

Edit: I corrected my solution to part (a). I had it flipped inside the parentheses.

ObiDonKenobi
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  • I’m voting to close this question because it is not about Astronomy as defined in the help center. However, it is exactly the kind of question that can get answered in Math SE, so hopefully it can be migrated there quickly. – uhoh Mar 07 '21 at 23:21
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    @B--rian "closing because belongs on site X..." is not a close reason. OP decides were to ask, community decides if on-topic or off-topic here. We can also recommend a better site, but that isn't a close reason. – uhoh Mar 07 '21 at 23:25
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    @ObiDonKenobi I see you've already asked one question in Math SE and there's a "better asked in Physics SE" comment there as well. Don't let all these comments worry you or feel bad about the close vote here. I've flagged the question recommending speedy migration to Math SE and added the fifth vote to speed the process along. Even though some astronomers certainly do this, it seems five users feel that it's not likely to receive an answer from this community based on past experience here. – uhoh Mar 07 '21 at 23:30
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    @uhoh I couldn't put it into better words than you: I believe that this question has the highest probability to be answered at math SE rather than here. I guess I misunderstood you when you said something like "vote for close so that we can migrate it". – B--rian Mar 08 '21 at 01:07
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    @B--rian the only decision "we" as a community makes is if it is on-topic or not. It probably looks like I'm splitting hairs here, but "we" don't migrate. That's purely a moderator's decision, and activity. I note this regularly so that new users reading through comments will also pick up on the limited scope of what users do and don't decide. – uhoh Mar 08 '21 at 01:15
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    Hello! I have submitted an edit converting some of your equations to MathJax. You can find a basic tutorial here – jng224 Mar 06 '21 at 15:53
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    I’m voting to close this question because this question belongs to math stackexchange. – B--rian Mar 06 '21 at 11:19
  • Thanks for the migration votes, I'm still learning how these sites work. As you noted, I attempted to ask another question over there, and was told there to ask that one here. Haha. I'll get the hang of it. – ObiDonKenobi Mar 08 '21 at 21:45

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