Evaluate the indicated integral: $$\int \frac { 3 x ^ { 4 } + 60 x ^ { 3 } + 21 } { x ^ { 5 } + 25 x ^ { 4 } + 35 x + 4 } d x$$
A) $\frac { 3 } { 5 } \ln \left| x ^ { 4 } + 20 x ^ { 3 } + 7 \right| + C$
B) $\frac { 3 } { 5 } \ln \left| x ^ { 5 } + 25 x ^ { 4 } + 28 x \right| + C$
C) $\frac { 5 } { 3 } \ln \left| x ^ { 5 } + 25 x ^ { 4 } + 35 x + 4 \right| + C$
D) $\frac { 3 } { 5 } \ln \left| x ^ { 5 } + 25 x ^ { 4 } + 35 x + 4 \right| + C$

Question

Quizplus · Accepted Answer

The integral is of the form $\int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + C$. Here, $f(x) = x^5 + 25x^4 + 35x + 4$ and $f'(x) = 5x^4 + 100x^3 + 35$. The given numerator $3x^4 + 60x^3 + 21$ is a multiple of $f'(x)$ by $\frac{3}{5}$, leading to the solution $\frac{3}{5} \ln|f(x)| + C$.

Evaluate the Indicated Integral ∫3x4+60x3+21x5+25x4+35x+4dx\int \frac { 3 x ^ { 4 } + 60 x ^ { 3 } + 21 } { x ^ { 5 } + 25 x ^ { 4 } + 35 x + 4 } d x∫x5+25x4+35x+43x4+60x3+21​dx

Evaluate the Indicated Integral $\int \frac { 3 x ^ { 4 } + 60 x ^ { 3 } + 21 } { x ^ { 5 } + 25 x ^ { 4 } + 35 x + 4 } d x$