Lim _ k 0 , ,1 k _0^k (1 + 2x) ^ 1 x dx - Vidyadip

MCQ

$\mathop {Lim}\limits_{k \to 0} \,\,\frac{1}{k}\int\limits_0^k {{{(1 + \sin 2x)}^{\frac{1}{x}}}dx} $

A
$2$
B
$1$
✓
$e^2$
D
non existent

✓

Answer

Correct option: C.

$e^2$

c
$l =$ $\mathop {Lim}\limits_{k \to 0} \,\,\frac{{\int\limits_0^k {{{(1 + \sin 2x)}^{\frac{1}{x}}}dx} }}{k}$
differentiating Using $ L’$ opital rule

$ l = $ $\mathop {Lim}\limits_{k\, \to \,0} \,{(1 + \sin 2k)^{\frac{1}{k}}}$ = ${e^{\mathop {Lim}\limits_{k \to 0} \,\frac{1}{k}(\sin 2k)}}$ = $e^2$

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 inverse of the function $\text{f}:\text{R}\rightarrow\{\text{x}\in\text{R}:\text{x}<1\}$ given by $\text{f(x)}=\frac{\text{e}^{\text{x}}-\text{e}^{-\text{x}}}{\text{e}^\text{x}+\text{e}^{-\text{x}}}$ is:

$\frac{1}{2}\log\frac{1+\text{x}}{1-\text{x}}$
$\frac{1}{2}\log\frac{2+\text{x}}{2-\text{x}}$
$\frac{1}{2}\log\frac{1-\text{x}}{1+\text{x}}$
$\text{None of these}$

Consider $f(x) = [x] + \sqrt {\left\{ X \right\}}$ where $[.]$ denotes greatest integer function and $\{.\}$ denotes fractional part function. Identify the correct statement-

If $A$ is a square matrix such that $A^2=A$, then $(I-A)^3+A$ is equal to

The vector equation of the line joining the points $i-2j+k$ and $-2j+3k$ is

The function $y = a(1 - \cos x)$ is maximum when $x = $

If $\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}, \vec{b}=-\hat{i}+2 \hat{j}+\hat{k}$ and $\vec{c}=3 \hat{i}+\hat{j}+2 \hat{k}$, then the value of $\vec{a} \cdot(\vec{b} \times \vec{c})$ is

The length of perpendicular from the origin to the plane which makes intercepts $\frac{1}{3},\frac{1}{4},\frac{1}{5}$ respectively on the coordinate axes is:

$\frac{1}{\sqrt[5]{2}}$
$\frac{1}{10}$
$\sqrt[5]{2}$
$5$

If $u = x{y^2}{\tan ^{ - 1}}\left( {{y \over x}} \right)$, then $x{u_x} + y{u_y} = $

Let $f$ be a real-valued function defined on the interval $(0, \infty)$ by $f(x)=\ln x+\int_0^x \sqrt{1+\sin t} d t$. Then which of the following statement(s) is (are) true?

$(A)$ $f^{\prime \prime}(x)$ exists for all $x \in(0, \infty)$

$(B)$ $f^{\prime}(x)$ exists for all $x \in(0, \infty)$ and $f^{\prime}$ is continuous on $(0, \infty)$, but not differentiable on $(0, \infty)$

$(C)$ there exists $\alpha>1$ such that $\left|f^{\prime}(x)\right|<|f(x)|$ for all $x \in(\alpha, \infty)$

$(D)$ there exists $\beta>0$ such that $|f(x)|+\left|f^{\prime}(x)\right| \leq \beta$ for all $x \in(0, \infty)$

Let the domain of the function $f(x)=\log _{4}\left(\log _{5}\left(\log _{3}\left(18 x-x^{2}-77\right)\right)\right)$ be $(a, b)$. Then the value of the integral $\int_{a}^{b} \frac{\sin ^{3} x}{\left(\sin ^{3} x+\sin ^{3}(a+b-x)\right)} d x$ is equal to $.....$