_ ^ x^n x ;dx = - Vidyadip

MCQ

$\int_{}^{} {{x^n}\log x\;dx = } $

A
$\frac{{{x^{n + 1}}}}{{n + 1}}\left\{ {\log x + \frac{1}{{n + 1}}} \right\} + c$
B
$\frac{{{x^{n + 1}}}}{{n + 1}}\left\{ {\log x + \frac{2}{{n + 1}}} \right\} + c$
C
$\frac{{{x^{n + 1}}}}{{n + 1}}\left\{ {2\log x - \frac{1}{{n + 1}}} \right\} + c$
✓
$\frac{{{x^{n + 1}}}}{{n + 1}}\left\{ {\log x - \frac{1}{{n + 1}}} \right\} + c$

✓

Answer

Correct option: D.

$\frac{{{x^{n + 1}}}}{{n + 1}}\left\{ {\log x - \frac{1}{{n + 1}}} \right\} + c$

d
(d)$\int_{}^{} {{x^n}\log x\,dx} = \log x\,.\,\frac{{{x^{n + 1}}}}{{n + 1}} - \int_{}^{} {\frac{{{x^{n + 1}}}}{{n + 1}}\,.\,\frac{1}{x}\,dx} $
$ = \frac{{{x^{n + 1}}}}{{n + 1}}\log x - \frac{{{x^{n + 1}}}}{{{{(n + 1)}^2}}} + c = \frac{{{x^{n + 1}}}}{{n + 1}}\left[ {\log x - \frac{1}{{n + 1}}} \right] + c.$

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