If I_n = ( x) ^n , ,dx , then I_n + n I_ n - 1 = - Vidyadip

MCQ

If ${I_n} = \int {{{(\log x)}^n}\,\,dx} ,$ then ${I_n} + n{I_{n - 1}} = $

✓
$x{(\log x)^n}$
B
${(x\log x)^n}$
C
${(\log x)^{n - 1}}$
D
$n{(\log x)^n}$

✓

Answer

Correct option: A.

$x{(\log x)^n}$

a
(a)${I_n} = \int {{{(\log x)}^n}dx} $ .....$(i)$
$\therefore {I_{n - 1}} = \int {{{(\log x)}^{n - 1}}dx} $.....$(ii)$
Now, ${I_n} = \int {{{(\log x)}^n}.\,dx} = {(\log x)^n}x - n\int {{{(\log x)}^{n - 1}}\frac{1}{x}x\,dx} $
$ = x{(\log x)^n} - n\int {{{(\log x)}^{n - 1}}dx} $
${I_n} = x{(\log x)^n} - n{I_{n - 1}}$; $\therefore {I_n} + n\,{I_{n - 1}} = x{(\log x)^n}$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $\alpha, \beta \in \mathbb{R}$ be such that $\lim _{x \rightarrow 0} \frac{x^2 \sin (\beta x)}{\alpha x-\sin x}=1$. Then $6(\alpha+\beta)$ equals

If ${z_1},{z_2},{z_3},{z_4}$ are the affixes of four points in the Argand plane and $z$ is the affix of a point such that $|z - {z_1}|\, = \,|z - {z_2}|\, = \,|z - {z_3}|\, = |z - {z_4}|$, then ${z_1},{z_2},{z_3},{z_4}$ are

The number of distinct primes dividing $12 !+13 !+14 !$ is

If $\alpha$ and $\beta $ are roots of ${x^2} - 3x + 1 = 0,$ then the equation whose roots are $\frac{1}{{\alpha - 2}},\frac{1}{{\beta - 2}}$ is

The total number of point of non-differentiability of $f\left( x \right) = \min \left\{ {\left| {\sin x} \right|,\left| {\cos x} \right|,\frac{1}{4}} \right\}$ in $(0, 2\pi)$ is

If $f(x) = x^5 - 5x^4 + 5x^3 -10$ has local maximum and minimum at $x = p$ and $x = q$, respectively, then $(p, q)$ equals

If $\overrightarrow {{{\left| c \right|}^2}} = 60$ and $\overrightarrow c \times \left( {\hat i + 2\hat j + 5\hat k} \right) = \overrightarrow 0 $, then a value of $\overrightarrow c .\left( { - 7\hat i + 2\hat j + 3\hat k} \right)$ is

A set $S$ contains $7$ elements. A non-empty subset $A$ of $S$ and an element $x$ of $S$ are chosen at random. Then the probability that $x \in A$ is

Two adjacent sides of a parallelogram $\mathrm{ABCD}$ are given by $\overrightarrow{A B}=2 \hat{i}+10 \hat{j}+11 \hat{k}$ and $\overrightarrow{A D}=-\hat{i}+2 \hat{j}+2 \hat{k}$. The side $\mathrm{AD}$ is rotated by an acute angle $\alpha$ in the plane of the parallelogram so that $\mathrm{AD}$ becomes $\mathrm{AD}^{\prime}$. If $\mathrm{AD}^{\prime}$ makes a right angle with the side $\mathrm{AB}$, then the cosine of the angle $\alpha$ is given by

A card is drawn from a pack of $52$ cards. If $A =$ card is of diamond, $B =$ card is an ace and $A \cap B = $ card is ace of diamond, then events $A$ and $B$ are