Integrate the function.
-$$\int _ { 0 } ^ { 2 } \frac { 81 d x } { \left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } }$$
A) $\sqrt { 77 } - 77$
B) $\frac { \sqrt { 77 } } { 77 }$
C) $\frac { 2 \sqrt { 77 } } { 77 }$
D) $77 ^ { 3 / 2 }$

Question

Quizplus · Accepted Answer

Use the substitution $x=9\sin t$ and $dx=9\cos tdt$. Then
$$\int _ { 0 } ^ { 2 } \frac { 81 d x } { \left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } }=\int _ { 0 } ^ { \frac{\pi}{2} } \frac { 81\cos t\, dt } { \left( 81 - 81\sin ^ { 2 } t \right) ^ { 3 / 2 } }$$
$$=\int _ { 0 } ^ { \frac{\pi}{2} } \frac { 1\, dt } { \cos^{2} t}=\int _ { 0 } ^ { \frac{\pi}{2} } \sec^{2}t\, dt=\left[\tan t\right] _ { 0 } ^ { \frac{\pi}{2} }=\frac { 2 \sqrt { 77 } } { 77 }$$

Integrate the Function ∫0281dx(81−x2)3/2\int _ { 0 } ^ { 2 } \frac { 81 d x } { \left( 81 - x ^ { 2 } \right) ^ { 3 / 2 } }∫02​(81−x2)3/281dx​

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