Download App

7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\frac{k}{{{n^2} + {k^2}}}} $is equals to
  • ✓
    $\frac{1}{2}\log 2$
  • B
    $log\ 2$
  • C
    $\pi /4$
  • D
    $\pi /2$
✓

Answer

Correct option: A.
$\frac{1}{2}\log 2$
a
(a) Let $I = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\,\frac{k}{{{n^2} + {k^2}}}} $

$ = \mathop {\lim }\limits_{n \to \infty } \,\sum\limits_{k = 1}^n {} \frac{1}{n}\frac{{\left( {\frac{k}{n}} \right)}}{{1 + {{\left( {\frac{k}{n}} \right)}^2}}}$

$I = \int\limits_0^1 {\frac{x}{{1 + {x^2}}}dx} $ 

$ = \frac{1}{2}[\log (1 + {x^2})]_{\,0}^{\,1}$$ = \frac{1}{2}\left[ {\log 2} \right]$.

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

More "M.C.Q (1 Marks)" from 7.2 definite integral→7.2 definite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Let A = {2, 3, 4, 5, ..., 17, 18}. Let $'\simeq'$ be the equivalence relation on A × A, cartesian product of A with itself, defined by $(\text{a, b})\simeq(\text{c, d)}$ if ad = bc. Then, the number of ordered pairs of the equivalence class of (3, 2) is:
  1. 4
  2. 5
  3. 6
  4. 7
→
Find the minor of the element 2 in the determinant $\triangle=\begin{bmatrix}1&9\\2&3\end{bmatrix}$?
  1. 3
  2. 9
  3. 1
  4. 2
→
The differential equation representing the family of ellipse having foci either on the $x-$ axis or on the $y-$ axis centre at the origin and passing through the point  $(0,3)$ is
→
Magnitudes of vectors $ \vec a,\vec b,\vec c$ are $3,4,5 $ respectively. If $\vec a$ and $ \vec b+\vec c, \vec b$ and $\vec c+ \vec a,\vec c $ and $\vec a + \vec b$ are mutually perpendicular, then magnitude of $|\vec a + \vec b + \vec c|$
→
The length of the perpendicular drawn from the point $(5, 4, -1)$ on the line $\frac{{x - 1}}{2} = \frac{y}{9} = \frac{z}{5}$ is
→
Evaluate $\begin{bmatrix}5&-4\\1&\sqrt{3}\end{bmatrix}$
  1. $4\sqrt{3}+4$
  2. $4\sqrt{3}+5$
  3. $5\sqrt{3}+4$
  4. $4\sqrt{3}-4$
→
Bag $I$ contains $3$ red,$4$ black and $3$ white balls and Bag $II$ contains $2$ red,$5$ black and $2$ white balls. One ball is transferred from Bag $I$ to Bag $II$ and then a ball is draw from Bag $II$. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red,is.
→
For positive integer $n$, define

$f(n)=n+\frac{16+5 n-3 n^2}{4 n+3 n^2}+\frac{32+n-3 n^2}{8 n+3 n^2}+\frac{48-3 n-3 n^2}{12 n+3 n^2}+\ldots+\frac{25 n-7 n^2}{7 n^2}$

Then, the value of $\lim _{ n \rightarrow \infty} f( n )$ is equal to

→
Square matrix ${[{a_{ij}}]_{n \times n}}$ will be an upper triangular matrix, if
→
Let $g(x)$ be the inverse of the function $f(x)$ and $f'(x) = {1 \over {1 + {x^3}}}$. Then $g'(x)$ is equal to
→