(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

See below.

2. The attempt at a solution

Hey there, apologies for doing so, but I don't know how to use latex on this board (it doesn't show me anything when I hit preview, so I had to make a picture for the sake of cleanness.

imgur.com/WzrAR.png

The first line is the original problem, with the proceeding lines being the attempt I made.

I have about four months before I transfer to a new university and figured I might as well self-study Calculus. I finished Calculus I without any trouble so I just started doing what I will eventually cover in Calculus II, but unfortunately my textbook offers only a mechanical way of doing this technique which leaves some doubt in my mind.

Is trigonometric substitution legal to do in any integral (regardless of whether it will actually help integrating) so long as you also substitute any related variables to the one you're substituting with? I've tried it out with simple indefinite integrals like ∫3x^2 dx but I'm unsure whether or not issues would occur if I substituted, for example, x = tan(θ) for ∫1/x due to the fact that 1/tan(θ) is undefined in an infinite amount of inputs while 1/x is undefined in only one.

Furthermore, I've looked over my work and I can't find any errors, so I don't know whether there's some concept I'm completely missing out on or there's some hidden error I just can't see.

Thanks for any input.

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# Integration involving trig. substitution

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