Mathematics

Mathematics

Problem

Find an antiderivative of x4−1x2+x.\dfrac{x^{4}-1}{x^{2}+x}. a) x33−x22+x−ln⁡∣x+1∣+C\dfrac{x^{3}}{3}-\dfrac{x^{2}}{2}+x-\ln|x+1|+C b) x33−x22+x+ln⁡∣x∣+C\dfrac{x^{3}}{3}-\dfrac{x^{2}}{2}+x+\ln|x|+C c) x33−x22+x−ln⁡∣x∣+C\dfrac{x^{3}}{3}-\dfrac{x^{2}}{2}+x-\ln|x|+C d) x33−x22+x−ln⁡∣x(x+1)∣+C\dfrac{x^{3}}{3}-\dfrac{x^{2}}{2}+x-\ln|x(x+1)|+C
Intuitive explanation

The key is to simplify this rational function before integrating. Use polynomial long division or factoring to rewrite the numerator in terms of the denominator and its derivative—this breaks the fraction into pieces you already know how to integrate. Once you've reduced it to simpler terms, you'll likely spot a mix of polynomial and logarithmic antiderivatives.

Intuitive explanation

The key is to simplify this rational function before integrating. Use polynomial long division or factoring to rewrite the numerator in terms of the denominator and its derivative—this breaks the fraction into pieces you already know how to integrate. Once you've reduced it to simpler terms, you'll likely spot a mix of polynomial and logarithmic antiderivatives.

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