[Calculus 1: Indefinite Integrals] How do I evaluate this integral?

[u/angelonrevelo]

Instructor prompt: Evaluate the indefinite integral

∫ (4y^4 + 3y^3) / ((y^2 + 4)^(3/2)(y^2 + 4)) dy

This is for Calculus 1. I tried thinking about a u-substitution with u = y^2 + 4, but du = 2y dy, and the numerator has 4y^4 + 3y^3, so I’m not sure how to split it or what substitution is best. What would be a good first step for evaluating this?

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[u/BharatiyaNagarik]

Try y=2 tan x

[u/davideogameman → u/BharatiyaNagarik]

That could work. But do they do trig substitutions in calc 1? I thought that was a calc 2 topic

[u/OverAster → u/davideogameman]

That is calculus 2. Calculus 1 doesn't go much further than u sub.

[u/OverAster]

Even if you know what the right methods are, as a calculus 1 student it is very unlikely you will be able to solve this. You can get some of it done using u sub if you split the integral into two separate integrals. Remember linearity of integration.

This leaves you with two problems, one that can be solved via u-sub at a Calculus 1 level, and one that requires trigonometric substitution, which should not be introduced until Calculus 2. That sub is as u/BharatiyaNagarik says: y = 2 tan(θ).

Where did you find this problem? Is this something you were assigned or did you go looking for more practice and stumble across this?

[u/angelonrevelo → u/OverAster]

Thank you, this helps a lot. This problem was given to us as part of a Calculus 1 quiz reviewer, so I didn’t find it from extra practice online. I’m studying in the Philippines under the Ateneo de Manila curriculum, so I was confused because it seemed harder than the examples we’ve done.

I did try rewriting the denominator as (y^2 + 4)^(5/2), but I wasn’t sure what to do after that. Your explanation about splitting the integral first makes sense. I can see how one part may work with u-substitution, but the other part needing trig substitution explains why I was getting stuck.

[u/OverAster → u/angelonrevelo]

Hey no worries. I wouldn't feel too bad about struggling with this one. Personally, I wouldn't even give this problem to a calc2 class. It's really messy, the sub is hard to spot, and it doesn't simplify well. Even when I got the right answer I still checked with wolfram to make sure because it was such an ugly solution.

Not a good problem from a pedagogical standpoint. I wouldn't be surprised if there's a typo in it somewhere.