Integration: Trig Substitution

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

[tex]\int \sqrt{x^2+c^2}dx[/tex]
[tex]\int \sqrt{x^2-c^2}dx[/tex]
[tex]\int \sqrt{c^2-x^2}dx[/tex]

 

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).

 

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).

 

Instead of the tangent-secans relation, you can also use the fact that [itex]\cosh ^2 x - \sinh ^2 x = 1[/itex]. (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.

 

Indeed, for example

 

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