Question

\(\int \frac{5x^{12}+6x^{11}}{(x^{6}+x+1)^{3}}dx=\)

• A. \(-\frac{x^{12}}{2(x^{6}+x+1)^{2}}+c\)
• B. \(-\frac{1}{2(x^{6}+x+1)^{2}}+c\)
• C. \(\frac{x^{12}}{2(x^{6}+x+1)^{2}}+c\)
• D. \(\frac{1}{2(x^{6}+x+1)^{2}}+c\)

Hint:

Divide numerator and denominator by x to the 18 so that the numerator becomes the derivative of the new base.

Answer 1

The Correct Option is C

Step 1: Guess the form
The numerator has degree 12 and the denominator has degree 18, so try $\frac{x^{12}}{2(x^{6}+x+1)^{2}}$.

Step 2: Differentiate
Using the quotient rule: $\frac{12x^{11}(x^{6}+x+1)-x^{12}\cdot2(6x^{5}+1)}{2(x^{6}+x+1)^{3}}$.

Step 3: Simplify
Numerator: $12x^{17}+12x^{12}+12x^{11}-12x^{17}-2x^{12}=10x^{12}+12x^{11}$.
Dividing by 2 gives $\frac{5x^{12}+6x^{11}}{(x^{6}+x+1)^{3}}$, the integrand.

Final Answer:
The third option differentiates back to the integrand. \[ \boxed{\text{(C) }\dfrac{x^{12}}{2(x^{6}+x+1)^{2}}+c} \]