Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula.
-$$\int \frac { d x } { \left( 25 - x ^ { 2 } \right) ^ { 2 } }$$
A) $\frac { 1 } { 10 } \ln \left| \frac { x + 5 } { x - 5 } \right| + C$
B) $\frac { x } { 50 \left( 25 - x ^ { 2 } \right) } + C$
C) $\frac { 1 } { 50 } \left( \frac { x } { 25 - x ^ { 2 } } + \frac { 1 } { 10 } \ln \left| \frac { x + 5 } { x - 5 } \right| \right) + C$
D) $\frac { 1 } { 50 } \left( \frac { x } { 25 - x ^ { 2 } } - \frac { 1 } { 10 } \ln \left| \frac { x + 5 } { x - 5 } \right| \right) + C$

Question

Quizplus · Accepted Answer

This integral can be solved by using a trigonometric substitution such as $x = 5\sin(\theta)$, which simplifies the integral into a form that can be integrated directly or through a reduction formula. The correct answer involves both a rational term and a logarithmic term, which matches the structure of option C, indicating the use of both trigonometric substitution and partial fraction decomposition or a similar method in the solution process.

Evaluate the Integral by Making a Substitution (Possibly Trigonometric) and Then