Use a table of integrals to evaluate the integral. $$\int \frac { \sqrt { 6 x + 5 } } { x ^ { 2 } } d x$$
A) $- \frac { \sqrt { 6 x + 5 } } { x } + \frac { 3 \sqrt { 5 } } { 5 } \ln \left| \frac { \sqrt { 6 x + 5 } - \sqrt { 5 } } { \sqrt { 6 x + 5 } + \sqrt { 5 } } \right| + C$
B) $- \frac { \sqrt { 6 x + 5 } } { x } + \frac { 6 \sqrt { 5 } } { 5 } \ln \left| \frac { \sqrt { 6 x + 5 } - \sqrt { 5 } } { \sqrt { 6 x + 5 } + \sqrt { 5 } } \right| + C$
C) $- \frac { 6 \sqrt { 5 } } { 5 } \ln \left| \frac { \sqrt { 6 x + 5 } - \sqrt { 5 } } { \sqrt { 6 x + 5 } + \sqrt { 5 } } \right| + C$
D) $- \frac { 3 \sqrt { 5 } } { 5 } \ln \left| \frac { \sqrt { 6 x + 5 } - \sqrt { 5 } } { \sqrt { 6 x + 5 } + \sqrt { 5 } } \right| + C$

Question

Quizplus · Accepted Answer

This integral requires a substitution to simplify, typically setting $u = \sqrt{6x + 5}$, which transforms the integral into a form that matches a standard integral formula. The correct answer includes both a term that results from the direct integration and a logarithmic term that comes from integrating a part that results from the substitution and partial fraction decomposition. The coefficients and the structure of the answer in choice A correctly reflect these steps.

Use a Table of Integrals to Evaluate the Integral ∫6x+5x2dx\int \frac { \sqrt { 6 x + 5 } } { x ^ { 2 } } d x∫x26x+5​​dx

Use a Table of Integrals to Evaluate the Integral $\int \frac { \sqrt { 6 x + 5 } } { x ^ { 2 } } d x$