$\displaystyle\int_2^3 \dfrac{x\,dx}{x^2+1}$ equals
A$\dfrac{1}{2}\log 2$
B$\log 2$
C$\dfrac{1}{2}\log\dfrac{5}{2}$
D$\dfrac{1}{2}\log\dfrac{3}{2}$
Answer & Solution
Correct answer: A. $\dfrac{1}{2}\log 2$
1. Put $t=x^2+1$, $dt=2x\,dx$, so $x\,dx=\tfrac12 dt$.
2. Antiderivative: $\dfrac{1}{2}\log(x^2+1)$.
3. Evaluate: $\dfrac{1}{2}[\log 10-\log 5]=\dfrac{1}{2}\log\dfrac{10}{5}=\dfrac{1}{2}\log 2$.
4. Option B drops the $\tfrac12$ factor.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.337_
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