Evaluate _1^2 3 x (9 x^2-1 ) d x - Vidyadip

Question

Evaluate $\int_1^2 \frac{3 x}{\left(9 x^2-1\right)} d x$

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Answer

$ \text { Let } I =\int_1^2 \frac{3 x}{\left(9 x^2-1\right)} d x$
$=3 \int_1^2 \frac{x}{9 x^2-1} d x $
Put $9 x^2-1=t$
$ \therefore 18 xd x= dt$
$\therefore xdx =\frac{1}{18} dt$
When $x=1, t=9(1)^2-1=8$
When $x=2, t=9(2)^2-1=35$
$ \therefore I =3 \int_8^{35} \frac{1}{ t } \cdot \frac{ dt }{18}$
$=\frac{1}{6} \int_8^{35} \frac{ dt }{ t } $
$=\frac{1}{6}[\log | t |]_8^{35}$
$=\frac{1}{6}(\log 35-\log 8)$
$\therefore I =\frac{1}{6} \log \left(\frac{35}{8}\right)$

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