Evaluate the integral.
-$$\int \frac { \sqrt { 25 x ^ { 2 } - 9 } } { x } d x$$
A) $\sqrt { x ^ { 2 } - 9 } - 3 \sec - 1 \left| \frac { x } { 3 } \right| + C$
B) $\ln \left| x + \sqrt { 25 x ^ { 2 } - 9 } \right| - \frac { \sqrt { 25 x ^ { 2 } - 9 } } { x } + C$
C) $\ln \left| x + \sqrt { x ^ { 2 } - 9 } \right| - \frac { \sqrt { x ^ { 2 } - 9 } } { x } + C$
D) $\sqrt { 25 x ^ { 2 } - 9 } - 3 \sec - 1 \left| \frac { 5 x } { 3 } \right| + C$

Question

Quizplus · Accepted Answer

The integral $\int \frac { \sqrt { 25 x ^ { 2 } - 9 } } { x } d x$ can be solved using trigonometric substitution. The correct answer involves recognizing the form $a^2x^2 - b^2$ under the square root, which suggests using a substitution related to secant. The answer involves the inverse secant function and the correct scaling factor for $x$ in the argument of the secant function, which is $\frac{5x}{3}$ in this context.

Evaluate the Integral ∫25x2−9xdx\int \frac { \sqrt { 25 x ^ { 2 } - 9 } } { x } d x∫x25x2−9​​dx

Evaluate the Integral $\int \frac { \sqrt { 25 x ^ { 2 } - 9 } } { x } d x$