The value of _1^ e^2 dx x (1 + x) ^2 is - Vidyadip

MCQ

The value of $\int_1^{{e^2}} {\frac{{dx}}{{x{{(1 + \ln x)}^2}}}} $ is

✓
$2/3$
B
$1/3$
C
$3/2$
D
$\ln 2$

✓

Answer

Correct option: A.

$2/3$

a
(a) $I = \int_1^{{e^2}} {\frac{{dx}}{{x{{(1 + \ln x)}^2}}}} $

Let $(1 + \ln x) = t$

==> $dt = \frac{1}{x}dx$

Now, when $x = 1 \to {e^2}$, then $t = 1 \to 3$

$\therefore $ $I = \int_1^3 {\frac{{dt}}{{{t^2}}} = \left[ {\frac{{ - 1}}{t}} \right]_1^3= - \left[ {\frac{1}{3} - 1} \right]} = \frac{2}{3}$.

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