Integrate the function.
-$$\int _ { 0 } ^ { 4 } \frac { 81 d x } { \left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } }$$
A) $65 ^ { 3 / 2 }$
B) $\sqrt { 65 } - 65$
C) $\frac { 4 \sqrt { 65 } } { 65 }$
D) $\frac { \sqrt { 65 } } { 65 }$

Question

Quizplus · Accepted Answer

This integral can be solved by recognizing it as a standard form related to the derivative of the inverse sine function, $\frac{d}{dx}(\sin^{-1}(x/a)) = \frac{1}{\sqrt{a^2 - x^2}}$, but with a cubic denominator and a slight adjustment for the constants involved. By substituting $x = 9\sin(\theta)$, we can simplify the integral and eventually find that the correct answer is $\frac { 4 \sqrt { 65 } } { 65 }$, which corresponds to choice C.

Integrate the Function ∫0481dx(81−x2)3/2\int _ { 0 } ^ { 4 } \frac { 81 d x } { \left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } }∫04​(81−x2)3/281dx​

Integrate the Function $\int _ { 0 } ^ { 4 } \frac { 81 d x } { \left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } }$