Evaluate : _2^4 x x^2+1 d x - Vidyadip

Question

Evaluate : $\int_2^4 \frac{x}{x^2+1} d x$

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Answer

$(a) :$ Let $I=\int_2^4 \frac{x}{x^2+1} d x$
Put $x^2+1=t \Rightarrow 2 x d x=d t$
$\Rightarrow x d x=\frac{1}{2} d t$
Also, $x=2$
$\Rightarrow t=5$ and $x=4$
$\Rightarrow t=17$
$\therefore I=\frac{1}{2} \int_5^{17} \frac{d t}{t}=\frac{1}{2}[\log t]_5^{17}$
$=\frac{1}{2}[\log 17-\log 5]=\frac{1}{2} \log \left(\frac{17}{5}\right)$

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