Question

what change of variables is suggested by an integral containing $sqrt{x^{2}-484}$?
choose the correct answer.
a. $x = 22sec\theta$
b. $x = 22\tan\theta$
c. $x = 484sec\theta$
d. $x = 22sin\theta$

Explanation:

Step1: Recall trig - substitution rules

For an integral with $\sqrt{a^{2}-x^{2}}$, the substitution $x = a\sin\theta$ is used. Here, if we assume the integral has $\sqrt{484 - x^{2}}$ (since $484=22^{2}$), and we want to get rid of the square - root. Using the trigonometric identity $\sin^{2}\theta+\cos^{2}\theta = 1$, when $x = 22\sin\theta$, then $\sqrt{484 - x^{2}}=\sqrt{484-484\sin^{2}\theta}=\sqrt{484(1 - \sin^{2}\theta)} = 22\cos\theta$.

Answer:

D. $x = 22\sin\theta$