Evaluate the indicated integral: $$\int \frac { 7 x ^ { 4 } + 56 x ^ { 3 } + 56 } { x ^ { 5 } + 10 x ^ { 4 } + 40 x + 4 } d x$$
A) $\frac { 7 } { 5 } \ln \left| x ^ { 5 } + 10 x ^ { 4 } + 40 x + 4 \right| + C$
B) $\frac { 5 } { 7 } \ln \left| x ^ { 5 } + 10 x ^ { 4 } + 40 x + 4 \right| + C$
C) $\frac { 7 } { 5 } \ln \left| x ^ { 4 } + 8 x ^ { 3 } + 8 \right| + C$
D) $\frac { 7 } { 5 } \ln \left| x ^ { 5 } + 10 x ^ { 4 } + 32 x \right| + C$

Question

Quizplus · Accepted Answer

The integral can be solved by recognizing that the numerator is the derivative of the denominator. The derivative of $x^5 + 10x^4 + 40x + 4$ is $5x^4 + 40x^3 + 40$, which, when multiplied by $\frac{7}{5}$, gives the numerator $7x^4 + 56x^3 + 56$. Therefore, the integral is of the form $\int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + C$, leading to the answer $\frac{7}{5} \ln|x^5 + 10x^4 + 40x + 4| + C$.

Evaluate the Indicated Integral ∫7x4+56x3+56x5+10x4+40x+4dx\int \frac { 7 x ^ { 4 } + 56 x ^ { 3 } + 56 } { x ^ { 5 } + 10 x ^ { 4 } + 40 x + 4 } d x∫x5+10x4+40x+47x4+56x3+56​dx

Evaluate the Indicated Integral $\int \frac { 7 x ^ { 4 } + 56 x ^ { 3 } + 56 } { x ^ { 5 } + 10 x ^ { 4 } + 40 x + 4 } d x$