# Integration: Trig Substitution

## Main Question or Discussion Point

When is th ebest time to use it and what are some good rules of thumb for it?

Usually when you an integral in one of these three forms:

$$\int \sqrt{x^2+c^2}dx$$
$$\int \sqrt{x^2-c^2}dx$$
$$\int \sqrt{c^2-x^2}dx$$

Tom Mattson
Staff Emeritus
Gold Member
It's much more general than that, though. The quadratic forms in those integrands need not appear under a radical, and they need not appear in the "numerator line" of an expression (IOW, they can be on the bottom).

HallsofIvy
Homework Helper
Generally speaking you should remember that
cos2x= 1- sin2x
tan2x= sec2x- 1 and
sec2x= 1+ tan2x

Any time you have 1- x2, x2- 1, or 1+ x2 or can reduce to (as, for example 9- x2) you might consider using a trig substitution (unless, of course, something simpler works).

TD
Homework Helper
Instead of the tangent-secans relation, you can also use the fact that $\cosh ^2 x - \sinh ^2 x = 1$. (cp the law with sin and cos, but here with a - instead of a +)

TD said:
Instead of the tangent-secans relation, you can also use the fact that $\cosh ^2 x - \sinh ^2 x = 1$. (cp the law with sin and cos, but here with a - instead of a +)
Yeah. At least for me, that has come up a lot in dealing with arc-length.

TD
Homework Helper
Indeed, for example HallsofIvy
Instead of the tangent-secans relation, you can also use the fact that $\cosh ^2 x - \sinh ^2 x = 1$. (cp the law with sin and cos, but here with a - instead of a +)