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{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 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

The integral $\int \frac{ e ^{3 \log _{e} 2 x }+5 e ^{2 \log _{ e } 2 x }}{ e ^{4 \log _{e} x }+5 e ^{3 \log _{e} x }-7 e ^{2 \log _{e} x }} dx , x > 0$, is equal to ....... .

(where $c$ is a constant of integration)

→
$\tan^{-1}1+\cos^{-1}\Big(\frac{-1}{2}\Big)+\sin^{-1}\Big(\frac{-1}{2}\Big)$
  1. $\frac{2\pi}{3}$
  2. $\frac{3\pi}{4}$
  3. $\frac{\pi}{2}$
  4. $6\pi$
→
If $a, b, c,$ are non zero complex numbers satisfying $a^2 + b^2 + c^2 = 0$ and $\left| {\begin{array}{*{20}{c}}
{{b^2} + {c^2}}&{ab}&{ac}\\
{ab}&{{c^2} + {a^2}}&{bc}\\
{ac}&{bc}&{{a^2} + {b^2}}
\end{array}} \right| = k{a^2}{b^2}{c^2},$ then $k$ is equal to
→
Choose the correct answer in Exercises:
The anti derivative of $\bigg(\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}}\bigg)$equals
  1. $\frac{1}{3}\text{x}^{\frac{1}{3}}+2\text{x}^{\frac{1}{2}}+\text{c}$
  2. $\frac{2}{3}\text{x}^{\frac{2}{3}}+\frac{1}{2}\text{x}^2+\text{c}$
  3. $\frac{2}{3}\text{x}^{\frac{3}{2}}+2\text{x}^{\frac{1}{2}}+\text{c}$
  4. $\frac{3}{2}\text{x}^{\frac{3}{2}}+\frac{1}{2}\text{x}^{\frac{1}{2}}+\text{c}$
→
The derivative of $\text{f(x)}=\int\limits^{\text{x}^3}_{\text{x}^2}\frac{1}{\log_{\text{e}}\text{t}}\text{ dt},(\text{x}>0),$ is:

  1. $\frac{1}{3\ln\text{x}}$

  2. $\frac{1}{3\ln\text{x}}-\frac{1}{2\ln\text{x}}$

  3.  $\big(\ln\text{x}\big)^{-1}\text{x}(\text{x}-1)$

  4. $\frac{3\text{x}^2}{\ln\text{x}}$

→
If $\frac{\sin ^{-1} x}{a}=\frac{\cos ^{-1} x}{b}=\frac{\tan ^{-1} y}{c} ; 0< x< 1,$ then the value of $\cos \left(\frac{\pi c }{ a + b }\right)$ is
→
The corner points of the feasible region determined by the system of linear constraints are (0, 10),(5, 5),(15, 15),(0, 20). Let z = px + qy where p, q > 0. Condition on p and q so that the maximum of z occurs at both the points (15, 15) and (0, 20) is __________:
→
Let $f:[0, \infty) \rightarrow[0, \infty)$ be defined as

$\mathrm{f}(\mathrm{x})= \int_{0}^{x}[y] \,d y$

Where $[x]$ is the greatest integer less than or equal to $x$. Which of the following is true?

→
If $\text{A}=\begin{bmatrix}1&-1\\2&-1\end{bmatrix},\text{B}=\begin{bmatrix}\text{a}&1\\\text{b}&-1\end {bmatrix}$ and (A + B)2 = A2 + B2, values of a and b are:
  1. a = 4, b = 1
  2. a = 1, b = 4
  3. a = 0, b = 4
  4. a = 2, b = 4
→
If $\text{S}=\begin{bmatrix}\text{a} & \text{b} \\ \text{c} & \text{d} \end{bmatrix},$ then adj A is:

  1. $\begin{bmatrix} -\text{d} & -\text{b} \\ -\text{c} & \text{a} \end{bmatrix}$

  2. $\begin{bmatrix} \text{d} & -\text{b} \\ -\text{c} & \text{a} \end{bmatrix}$

  3. $\begin{bmatrix} \text{d} & \text{b} \\ \text{c} & \text{a} \end{bmatrix}$

  4. $\begin{bmatrix} \text{d} & \text{c} \\ \text{b} & \text{a} \end{bmatrix}$

→