KVPY SX SB Exam 2013 Question Paper

© examsiri.com

Question : 88 of 120

Marks: +1, -0

Let n be a positive integer. For a real number x, let [x] denote the largest integer not exceeding x and

\{x\}=x-[x].

Then

∫↖{n+1}↙{1}(\{x\})^[x]/[x]dx

is equal to

$\log_e(n)$
$1/{n+1}$
$n/{n+1}$
$1+1/2+...+1/n$

Solution:

∫↖{n+1}↙{1}\{x\}^[x]/[x]dx

=∫↖{2}↙{1}\{x\}^[x]/[x]dx

+∫↖{3}↙{2}\{x\}^[x]/[x]dx

+...+∫↖{n+1}↙{n}\{x\}^[x]/[x]dx

=∑↖{n}↙{r=1}∫↖{r+1}↙{r}\{x\}^[x]/[x]dx

=∑↖{n}↙{r=1}∫↖{r+1}↙{r} (x-r)^r/rdx

=∑↖{n}↙{r=1}∫↖{r+1}↙{r} [(x-r)^{r+1}/{r(r+1)}]^{r+1}_r

=∑↖{n}↙{r=1} 1/{r(r+1)}=n/{n+1}

© examsiri.com

Go to Question:

Prev Question

Next Question