# Integration: Trig Substitution

1. Sep 25, 2005

### badtwistoffate

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

2. Sep 25, 2005

### Jameson

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

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

3. Sep 25, 2005

### Tom Mattson

Staff Emeritus
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).

4. Sep 26, 2005

### HallsofIvy

Staff Emeritus
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).

5. Sep 26, 2005

### TD

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 +)

6. Sep 26, 2005

### amcavoy

Yeah. At least for me, that has come up a lot in dealing with arc-length.

7. Sep 26, 2005

### TD

Indeed, for example

8. Sep 26, 2005

### HallsofIvy

Staff Emeritus
Yeah, but that wouldn't be a trig substitution, would it!

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