evaluate the following integral using trigonometric substitution.\n int\frac{dx}{(289 + x^{2})^{\frac{3}{2}}}…

evaluate the following integral using trigonometric substitution.\n int\frac{dx}{(289 + x^{2})^{\frac{3}{2}}}\nwhat substitution will be the most helpful for evaluating this integral?\na. (x = 17\tan\theta)\nb. (x = 17sin\theta)\nc. (x = 17sec\theta)

evaluate the following integral using trigonometric substitution.\n int\frac{dx}{(289 + x^{2})^{\frac{3}{2}}}\nwhat substitution will be the most helpful for evaluating this integral?\na. (x = 17\tan\theta)\nb. (x = 17sin\theta)\nc. (x = 17sec\theta)

Answer

Explanation:

Step1: Recall trig - substitution rules

For an integral of the form $\int\frac{dx}{(a^{2}+x^{2})^n}$, the substitution $x = a\tan\theta$ is useful. Here $a^{2}=289$, so $a = 17$ and $n=\frac{3}{2}$.

Step2: Check the substitution

If $x = 17\tan\theta$, then $dx=17\sec^{2}\theta d\theta$ and $289 + x^{2}=289+289\tan^{2}\theta=289(1 + \tan^{2}\theta)=289\sec^{2}\theta$.

Answer:

A. $x = 17\tan\theta$