Find the indefinite integral. $\int \frac { x ^ { 2 } + 16 x + 6 } { x ^ { 3 } + 24 x ^ { 2 } + 18 x - 1 } d x$
A) $\frac { 1 } { 3 } \ln \left| x ^ { 3 } + 24 x ^ { 2 } + 18 x - 1 \right| + C$
B) $\ln \left| x ^ { 3 } + 24 x ^ { 2 } + 18 x - 1 \right| + C$
C) $\frac { 1 } { 3 } \ln \sqrt [ 3 ] { \left| x ^ { 3 } + 24 x ^ { 2 } + 18 x - 1 \right| } + C$
D) $3 \ln \left| x ^ { 2 } + 16 x + 6 \right| + C$
E) $\ln \sqrt { \left| x ^ { 3 } + 24 x ^ { 2 } + 18 x - 1 \right| } + C$

Question

Quizplus · Accepted Answer

The integral is of the form $\int \frac{f'(x)}{f(x)} dx$, where $f(x) = x^3 + 24x^2 + 18x - 1$. The derivative of $f(x)$ is $f'(x) = 3x^2 + 48x + 18$, which is a constant multiple of the numerator. Thus, the integral simplifies to $\frac{1}{3} \ln |f(x)| + C$.

Find the Indefinite Integral ∫x2+16x+6x3+24x2+18x−1dx\int \frac { x ^ { 2 } + 16 x + 6 } { x ^ { 3 } + 24 x ^ { 2 } + 18 x - 1 } d x∫x3+24x2+18x−1x2+16x+6​dx

Find the Indefinite Integral $\int \frac { x ^ { 2 } + 16 x + 6 } { x ^ { 3 } + 24 x ^ { 2 } + 18 x - 1 } d x$