_ ,1 ^ ,x x^2 x ,dx = - Vidyadip

MCQ

$\int_{\,1}^{\,x} {\frac{{\log {x^2}}}{x}\,dx = } $

✓
${(\log x)^2}$
B
$\frac{1}{2}{(\log x)^2}$
C
$\frac{{\log {x^2}}}{2}$
D
None of these

✓

Answer

Correct option: A.

${(\log x)^2}$

a
(a) $I = \int_1^x {\frac{{2\log x}}{x}dx} $

Let $\log x = t$

==> $\frac{{dx}}{x} = dt$

$\therefore I = 2\int_0^{\log x} {t\,dt = 2\,\left[ {\frac{{{t^2}}}{2}} \right]} _0^{\log x} = {(\log x)^2}$.

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