Question

What are the allowable forms of an expression for trigonometric substitution?

I know that
\sqrt{x^2 - a^2}
\sqrt{a^2 - x^2}
\sqrt{x^2 + a^2}

^- these count, but what are others?

Answer 1

ArcSin=> 1/root(1-u^2) (du/dx)
ArcTan=> 1/(1+u^2) (du/dx)
ArcCos=> -1/root(1-u^2) (du/dx)
ArcCot=> -1/(1+u^2) (du/dx)

Answer 2

\[\sqrt{a^2-b^2x^2}\implies x=\frac ab\sin\theta\]\[\sqrt{b^2x^2-a^2}\implies x=\frac ab\sec\theta\]\[\sqrt{b^2x^2+a^2}\implies x=\frac ab\tan\theta\]those are the only ones you really need to know

Answer 3

Does a^2 + x^2 <-- this count?

Answer 4

yes, that is my third formula above with b=1