1. ## Trig Substitution Strategy

When the variable under the square root is not a square, then what is the general strategy? Any example?

If the variable is a square then

$\displaystyle \sqrt{a - x^{2}}$ then $\displaystyle a\sin\theta$

$\displaystyle \sqrt{a + x^{2}}$ then $\displaystyle a\tan\theta$

$\displaystyle \sqrt{x^{2} - a}$ then $\displaystyle a\sec\theta$

2. ## Re: Trig Substitution Strategy

If it's a linear function under the square root, just substitute u for that entire linear function.

3. ## Re: Trig Substitution Strategy

Originally Posted by Jason76
When the variable under the square root is not a square, then what is the general strategy? Any example?

If the variable is a square then

$\displaystyle \sqrt{a - x^{2}}$ then $\displaystyle a\sin\theta$

$\displaystyle \sqrt{a + x^{2}}$ then $\displaystyle a\tan\theta$

$\displaystyle \sqrt{x^{2} - a}$ then $\displaystyle a\sec\theta$
Wrong, I'm afraid:

$\displaystyle \sqrt{a^2 - x^{2}}$ then $\displaystyle a\sin\theta$

$\displaystyle \sqrt{a^2 + x^{2}}$ then $\displaystyle a\tan\theta$

$\displaystyle \sqrt{x^{2} - a^2}$ then $\displaystyle a\sec\theta$

is correct.

One good reason for using $\displaystyle a^2$ rather than $\displaystyle a$ is because then it's obvious what the sign is. $\displaystyle a^2$ is always positive.

You should be able to find all this stuff in a basic primer on calculus: there is invariably a list of "useful" substitutions if the book you're using is worth a damn. If not, then find them on line.