_a^b x x d x= - Vidyadip

MCQ

$\int_a^b \frac{\log x}{x} d x=$

A
$\log \left(\frac{\log b}{\log a}\right)$
B
$\log (a b) \log \left(\frac{b}{a}\right)$
✓
$\frac{1}{2} \log (a b) \log \left(\frac{b}{a}\right)$
D
$\frac{1}{2} \log ( ab ) \log \left(\frac{ a }{ b }\right)$

✓

Answer

Correct option: C.

$\frac{1}{2} \log (a b) \log \left(\frac{b}{a}\right)$

(C)
Let $I =\int_{ a }^{ b } \frac{1}{x} \log x d x$
$\begin{array}{l}\Rightarrow I =[\log x \log x]_{ a }^{ b }-\int_{ a }^{ b } \frac{1}{x} \log x d x \\ \Rightarrow 2 I =\left[(\log x)^2\right]_{ a }^{ b }\end{array}$
$\Rightarrow I =\frac{1}{2}\left[(\log b)^2-(\log a )^2\right]$
$\begin{array}{l}=\frac{1}{2}[(\log b+\log a)(\log b-\log a)] \\ =\frac{1}{2} \log (a b) \log \left(\frac{b}{a}\right)\end{array}$

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