Find the indefinite integral. $$\int \frac { d x } { \sqrt { - x ^ { 2 } - 18 x } }$$
A) $\arcsin \left( \frac { x + 18 } { 18 } \right) + C$
B) $\arcsin \left( \frac { x + 9 } { 9 } \right) + C$
C) $\arcsin \left( \frac { x - 18 } { 18 } \right) + C$
D) $\arcsin \left( \frac { x - 9 } { 9 } \right) + C$
E) $\arcsin \left( \frac { x + 9 } { 18 } \right) + C$

Question

Quizplus · Accepted Answer

To solve the integral $\int \frac { d x } { \sqrt { - x ^ { 2 } - 18 x } }$, we first complete the square inside the square root to get it in a form suitable for a trigonometric substitution. The expression inside the square root is $-x^2 - 18x$. Completing the square, we get $- (x^2 + 18x + 81) + 81$, which simplifies to $- (x + 9)^2 + 81$. Factoring out the negative sign and rearranging gives us $\sqrt{81 - (x + 9)^2}$. This suggests a substitution with a trigonometric identity, leading to the form $\frac{1}{9}\sqrt{81 - (x + 9)^2}$, which resembles the derivative of $\arcsin\left(\frac{x + 9}{9}\right)$, hence the correct answer is $\arcsin \left( \frac { x + 9 } { 9 } \right) + C$.

Find the Indefinite Integral ∫dx−x2−18x\int \frac { d x } { \sqrt { - x ^ { 2 } - 18 x } }∫−x2−18x​dx​

Find the Indefinite Integral $\int \frac { d x } { \sqrt { - x ^ { 2 } - 18 x } }$