allery$\int \dfrac{x\,dx}{(x-1)(x-2)}$ equals
A$\log\left|\dfrac{(x-1)^2}{x-2}\right|+C$
B$\log\left|\dfrac{(x-2)^2}{x-1}\right|+C$
C$\log\left|\left(\dfrac{x-1}{x-2}\right)^2\right|+C$
D$\log|(x-1)(x-2)|+C$
Answer & Solution
Correct answer: B. $\log\left|\dfrac{(x-2)^2}{x-1}\right|+C$
1. Partial fractions: $\dfrac{x}{(x-1)(x-2)}=\dfrac{A}{x-1}+\dfrac{B}{x-2}$.
2. $x=1$: $1=A(-1)$ so $A=-1$. $x=2$: $2=B(1)$ so $B=2$.
3. Integrate: $-\log|x-1|+2\log|x-2|$.
4. Combine: $\log\left|\dfrac{(x-2)^2}{x-1}\right|+C$. Option A swaps the powers.
_Source: syllabus Mathematics Ch 7 "Integrals", p.322_
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