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

MCQ

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

✓
$\frac{1}{2} \log \left(\frac{17}{5}\right)$
B
$\frac{1}{2} \log \left(\frac{5}{17}\right)$
C
$\log \left(\frac{17}{5}\right)$
D
$\log \left(\frac{5}{17}\right)$

✓

Answer

Correct option: A.

$\frac{1}{2} \log \left(\frac{17}{5}\right)$

(a) : Let $I=\int_2^4 \frac{x}{x^2+1} d x$
Put $x^2+1=t \Rightarrow 2 x d x=d t \quad \Rightarrow \quad x d x=\frac{1}{2} d t$
Also, $x=2 \Rightarrow t=5$ and $x=4 \Rightarrow t=17$
$
\begin{array}{l}
\therefore \quad 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)
\end{array}
$

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