allery$\int \dfrac{x^3-1}{x^2}\,dx$ equals
A$\dfrac{x^2}{2}+\log|x|+C$
B$\dfrac{x^2}{2}-\dfrac{1}{x}+C$
C$\dfrac{x^3}{3}+\dfrac{1}{x}+C$
D$\dfrac{x^2}{2}+\dfrac{1}{x}+C$
Answer & Solution
Correct answer: D. $\dfrac{x^2}{2}+\dfrac{1}{x}+C$
1. Split: $\dfrac{x^3-1}{x^2}=x-x^{-2}$.
2. $\int x\,dx=\dfrac{x^2}{2}$.
3. $\int x^{-2}\,dx=\dfrac{x^{-1}}{-1}=-\dfrac{1}{x}$, so subtracting gives $+\dfrac{1}{x}$.
4. Total: $\dfrac{x^2}{2}+\dfrac{1}{x}+C$. Option B drops the sign on the second term.
_Source: syllabus Mathematics Ch 7 "Integrals", p.293_
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