Trigonometric, Half-Angle, and Hyperbolic Substitutions

All flashcards

Which radical form is commonly simplified by setting $x=a\sin\theta$?

A radical containing $a^2-x^2$, especially $\sqrt{a^2-x^2}$.

What is the differential when $x=a\sin\theta$?

Differentiating gives $dx=a\cos\theta\,d\theta$.

How does $x=a\sin\theta$ simplify $\sqrt{a^2-x^2}$?

It gives $a\cos\theta$ on the principal arcsine range when $a>0$.

What is an antiderivative of $1/\sqrt{a^2-x^2}$?

$\arcsin(x/a)+C$, for $a>0$ and $|x|<a$.

How are bounds converted under $x=a\sin\theta$?

Use $\theta=\arcsin(x/a)$ with angles in $[-\pi/2,\pi/2]$.

Quiz questions

1. Which substitution is standard for an integrand containing $\sqrt{a^2-x^2}$, with $a>0$?Answer: $x=a\sin\theta$
2. Which identity simplifies $1-\sin^2\theta$?Answer: $1-\sin^2\theta=\cos^2\theta$
3. If $x=a\sin\theta$, what is $dx$?Answer: $dx=a\cos\theta\,d\theta$
4. Why is $-\pi/2\leq\theta\leq\pi/2$ useful when simplifying $\sqrt{a^2-x^2}$ after $x=a\sin\theta$?Answer: Cosine is nonnegative, so $\sqrt{\cos^2\theta}=\cos\theta$.
5. Under $x=a\sin\theta$, what are the transformed bounds when $x$ goes from $0$ to $a/2$, for $a>0$?Answer: $\theta$ goes from $0$ to $\pi/6$.