Integral Calculus-1
Integral Calculus-1
Allen Star Batch
Grade 12
Question:
The derivative of $x^4 + x^{-5}$ is $-\left(4x^{-5} + 5x^{-6}\right)$. So, $$\int \frac{5x^3 + 4x^5}{\left(x^5 + x + 1\right)^2} dx =$$
$$x^5 + x + 1 + C$$
$$\frac{1}{x^5 + x + 1} + C$$
$$x^4 + x^{-5} + C$$
$$\frac{x^5}{x^5 + x + 1} + C$$
Step-by-Step Solution
Key Concept: Dividing by a high power of $x$ converts a difficult rational integral into a form that may be recognized or more easily substituted.
Divide both numerator and denominator by $x^{10}$ to get $\int \frac{5x^{-5} + 4x^{-6}}{(1 + x^{-4} + x^{-5})^2}dx$. This integral is standard and can be solved through recognition or substitution techniques applicable to reciprocal polynomial forms.
Correct Answer: 4