I see why the trig substitution leads to a simple integral, I just don't get why we can set x equal to any arbitrary function in any coordinate system. The comment below my post says it's due to the chain rule, but $\theta$ doesn't move in the same way x does, and trig functions repeat while most other functions don't.

Jan 25, 2015 · Without reference to $\theta$ substitution two curves can be drawn one above x-axis and another below x-axis. For the former, area under curve is > 0, and for latter, < 0. Share

It is hard to visualize the bounds of the substitution that will keep it positive but I think that is something I can just memorize from a table. So this is similar to u substitution except that I am not using a single variable but expressing x in the form of a trig function. How does this not change the value of the problem?

It is challenging to actually say it is "easier," since there is a strict algorithm for handling integrals of powers of $\sec x$ and for more general products of trig functions. So is it really easier to avoid it? I once wrote a blog post on trig sub vs. hyperbolic trig sub vs. Euler sub. I perform one integral using all three techniques to see ...

Oct 24, 2021 · From what I've gathered, every text book says we can do trig substitution with either sin, sec or tan. Some other examples use tan(x/2), but so far I've never encountered cosine substitution. Is th...

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Trig substitution using reference triangles. Related. 4. Which angle to pick for trigonometric ...

Understanding less trivial integration by trig substitution. 0. Integral using trig substitution. 1.

The idea here is that you want to simplify the radical using an identity like $1-\cos^2t=\sin^2t$ or $1+\tan^2t=\sec^2t$.

More to what you asked, any x outside the $\pm a$ range is complex, so having a trig substitution function that is "bounded" is totally appropriate. If you're doing complex integration, have bounds greater than $\pm a$, then you can extend the values for inverse trig functions, obviously they will involve complex numbers.