Evaluate: _ 1 ^ 2 3 x 9 x^ 2 -1 d x - Vidyadip

Question

Evaluate: $\int_{1}^{2} \frac{3 x}{9 x^{2}-1} d x$

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Answer

Put $9 x^{2}-1=t \Rightarrow 18 \mathrm{xdx}=\mathrm{dt} \Rightarrow 3 \mathrm{xdx}=\frac{1}{6} \mathrm{dt}$.
When $\mathrm{x}=1, \mathrm{t}=9.1^{2}-1=8$ and when $\mathrm{x}=2, \mathrm{t}=9.2^{2}-1=35$.
$\therefore \mathrm{I}=\frac{1}{6} \int_{8}^{35} \frac{1}{t} d t=\frac{1}{6}[\log |t|]_{8}^{35}=\frac{1}{6}(\log 35-\log 8)=\frac{1}{6} \log \frac{35}{8}$.

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