Soluion: This is also almost identical to the above examples, but there’s one glaring difference: You don’t This is considerably harder than the integral obtained in the first example but it’s not undoable, and if you’re clever with trig identities, you can get an answer here (which is more than you’d have gotten without Hence, the integral becomes: ∫Īt first glance, this probably appears no better, but rewriting tan and sec in terms of sin and cos tells us that ∫ Again, no u-substitution will work, and even though we have no square roots, we can use trig sub with Soluion: This is almost identical to the first example.

This can be integrated directly using a clever trick, but should probably instead be considered an integral you should know. X2 − 32, so letting x = 3 sec θ and dx = 3 sec θ tan θ dθ transforms the square root intoĩ tan2 θ = 3 tan θ. Soluion: Here, no u-substitution will work, and so we use trig sub. You should only do so if no other technique (e.g., u-substitution) works. In particular, if you have an integrand that looks like an expression inside the square roots shown in the above table, then you can use trig substitution.

As we saw in class, you can use trig substitution even when you don’t have square roots. ◦ Let x = a sec θ for 0 ≤ θ < π/2 (choose this if x ≥ a) or π ≤ θ < 3π/2 (choose this ifīecause integrals involving square roots are hard, and as the above table shows, using trig substitution can be a method for getting rid of square roots. There are three main forms of trig substitution you should know: Trig substitution is a somewhat-confusing technique which, despite seeming arbitrary, esoteric, and complicated (at best), is pretty useful for solving integrals for which no other technique we’ve learned thus far will work.

Trig integral identities cheat sheet