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$\int \dfrac{dx}{(x+1)(x+2)}$ equals

A$\log\left|\dfrac{x+2}{x+1}\right|+C$
B$\log\left|\dfrac{x+1}{x+2}\right|+C$
C$\log|(x+1)(x+2)|+C$
D$\dfrac{1}{(x+1)(x+2)}+C$
Answer & Solution
Correct answer: B. $\log\left|\dfrac{x+1}{x+2}\right|+C$
1. Partial fractions: $\dfrac{1}{(x+1)(x+2)}=\dfrac{A}{x+1}+\dfrac{B}{x+2}$ gives $A=1,\ B=-1$. 2. So integrand $=\dfrac{1}{x+1}-\dfrac{1}{x+2}$. 3. Integrate: $\log|x+1|-\log|x+2|$. 4. Combine: $\log\left|\dfrac{x+1}{x+2}\right|+C$. Option A inverts the ratio. _Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.317_
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