Lim _ n , , _ k = 1 ^n n n^2 + k^2 x^2 , x > 0 is equal to - Vidyadip

MCQ

$\mathop {Lim}\limits_{n \to \infty } \,\,\sum\limits_{k = 1}^n {\frac{n}{{{n^2} + {k^2}{x^2}}}} $,$ x > 0$ is equal to

A
$x tan^{-1}(x)$
B
$tan^{-1}(x)$
✓
$\frac{{{{\tan }^{ - 1}}(x)}}{x}$
D
$\frac{{{{\tan }^{ - 1}}(x)}}{{{x^2}}}$

✓

Answer

Correct option: C.

$\frac{{{{\tan }^{ - 1}}(x)}}{x}$

c
$T_{n}=\frac{n}{n^{2}+k^{2} x^{2}}=\frac{1}{n\left[1+(k / n)^{2} x^{2}\right]}$

$S=\frac{1}{n} \sum_{k=1}^{n} \frac{1}{1+\underbrace{(k / n)^{2} x^{2}}}_{1}=\int_{0}^{1} \frac{d t}{1+t^{2} x^{2}}$

$=\frac{1}{x^{2}} \int_{0}^{1} \frac{d t}{t^{2}+\left(1 / x^{2}\right)}=\left[\frac{1}{x} \tan ^{-1}(t x)\right]_{0}^{1}=\frac{\tan ^{-1}(x)}{x}$

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 STD 12 - 7.2 definite integral→STD 12 - 7.2 definite integral — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

If the function $f(x)\, = \left\{ {\begin{array}{*{20}{c}}{ - x,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x < 1\,\,\,\,}\\{a + {{\cos }^{ - 1}}(x + b),\,\,\,\,\,\,\,\,\,1 \le x \le 2} \end{array}} \right.$ is differentiable at $x = 1 ,$ then $\frac {a}{b}$ is equal to

The integrating factor of the differential equation $\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=2\text{x}^{2}.$

Let $f : R \rightarrow R$ be given by $f(x) = [x^2] + [x + 1] - 3$ where $[x]$ denotes the greatest integer less than or equal to $x.$ Then, $f(x)$ is:

Which of the following is correct?

The direction cosines of a line which is equally inclined to axes, is given by:

The polynomial $f (x)$ satisfies the condition $f (x + 1) = x^2 + 4x.$ The area enclosed by $y = f (x - 1)$ and the curve $x^2 + y = 0,$ is

If $x = {t^2}$, $y = {t^3}$, then ${{{d^2}y} \over {d{x^2}}} =$

Let $\vec a,\,\vec b,$ and $\vec c$ be three unit vectors, out of which vectors $\vec b$ and $\vec c$ are non-parallel. If $\alpha $ and $\beta $ are the angles which vector $\vec a$ makes with vectors $\vec b$ and $\vec c$ respectively and $\vec a\,\, \times \,\,(\vec b\,\, \times \,\,\vec c)\,\, = \,\,\frac{1}{2}\,\,\vec b,$ then $\left| {\alpha - \beta } \right|$ is equal to .............. $^o$

Two points are randomly chosen on the circumference of a circle of radius $r$. The probability that the distance between the two points is at least $r$ is equal to

Consider the binary operation $\times$ on $Q$ defind by $a \times b = a + 12b + ab$ for $a, b \in Q.$ Find $2\times\frac{1}{3}$