The integral _ 0 ^ 1 1 _ 7 ^ [1 x ] d x= where [.] denotes the gr… - Vidyadip

MCQ

The integral $\int_{0}^{1} \frac{1}{{ }_{7}^{\left[\frac{1}{x}\right]}} d x=$ where [.] denotes the greatest integer function is equal to

✓
$1+6 \log _{e}\left(\frac{6}{7}\right)$
B
$1-6 \log _{e}\left(\frac{6}{7}\right)$
C
$\log _{e}\left(\frac{7}{6}\right)$
D
$1-7 \log _{ e }\left(\frac{6}{7}\right)$

✓

Answer

Correct option: A.

$1+6 \log _{e}\left(\frac{6}{7}\right)$

a
$\int\limits_{0}^{1} \frac{1}{7\left[\frac{1}{x}\right]} dx =-\int\limits_{1}^{0} \frac{1}{\left.7^{\left[\frac{1}{x}\right.}\right]} dx$

$=(-1)\left[\int\limits_{1}^{1 / 2} \frac{1}{7} dx +\int\limits_{1 / 2}^{1 / 3} \frac{1}{7^{2}} dx +\int\limits_{1 / 3}^{1 / 4} \frac{1}{7^{3}} dx +\ldots \ldots \infty\right]$

$=\left(\frac{1}{7}+\frac{1}{2 \cdot 7^{2}}+\frac{1}{3 \cdot 7^{3}}+\ldots \infty\right)-\left(\frac{1}{7 \cdot 2}+\frac{1}{7^{2} \cdot 3}+\frac{1}{7^{2} \cdot 4} \ldots \infty\right)$

$=-\ln \left(1-\frac{1}{7}\right)-7\left(\frac{1}{7^{2} \cdot 2}+\frac{1}{7^{3} \cdot 3}+\frac{1}{7^{4} \cdot 4}+\ldots . \ldots\right)$

$= {\left[\ln \frac{6}{7}-7\left(-\ln \left(1-\frac{1}{7}\right)-\frac{1}{7}\right)\right.}$

$=6 \ln \frac{6}{7}+1$

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 $f:(-1,1) \rightarrow$ IR be such that $f(\cos 4 \theta)=\frac{2}{2-\sec ^2 \theta}$ for $\theta \in\left(0, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$. Then the value(s) of $f\left(\frac{1}{3}\right)$ is (are)

$(A)$ $1-\sqrt{\frac{3}{2}}$ $(B)$ $1+\sqrt{\frac{3}{2}}$ $(C)$ $1-\sqrt{\frac{2}{3}}$ $(D)$ $1+\sqrt{\frac{2}{3}}$

Let $\text{f(x)}=\frac{\text{x}-1}{\text{x}+1},$ then f(f(x)) is:

$\frac{1}{\text{x}}$
$-\frac{1}{\text{x}}$
$\frac{1}{\text{x}+1}$
$\frac{1}{\text{x}-1}$

The given table shows the number of cars manufactured in four different colours on a particular day. Study it carefully and answer the question.

	Number of cars manufactured
Colour	Vento	Creta	Wagonr
Red	65	88	93
White	54	42	80
Black	66	52	88
Sliver	37	49	74

Which car was twice the number of silver Vento?

Silver WagonR
Red WagonR
Red Vento
White Creta

If $y=\log _7(\log x)$, then find $\frac{d y}{d x}$.

For each positive integer $n$, defined $f_n(x)=$ minimum $\left(\frac{x^n}{n !}, \frac{(1-x)^n}{n !}\right)$, for $0 \leq x \leq 1 .$ Let $I_n=\int \limits_0^1 f_n(x) d x, n \geq 1$. Then, $\sum \limits_{n=1}^{\infty} I_n$ is equal to

If $A=\left[\begin{array}{cc}8 & 0 \\ 4 & -2 \\ 3 & 6\end{array}\right]$ and $B=\left[\begin{array}{cc}2 & -2 \\ 4 & 2 \\ -5 & 1\end{array}\right],$ then find the matrix $X$, such that $2 \mathrm{A}+3 \mathrm{X}=5 \mathrm{B}$.

A coin is tossed 10 times. The probability of getting exactly six heads is:

$\frac{512}{513}$
$\frac{105}{512}$
$\frac{100}{153}$
$\text{ }^{10}\text{C}_6$

Area of the region enclosed between the curves $x = y^2 -1$ and $x = |y|$ $\sqrt {1 - {y^2}} $ is

$\int \frac{x+3}{(x+4)^2} e^x d x$

Let $\alpha, \beta$ and $\gamma$ be real numbers. consider the following system of linear equations

$x+2 y+z=7$

$x+\alpha z=11$

$2 x-3 y+\beta z=\gamma$

Match each entry in List - $I$ to the correct entries in List-$II$

List - $I$	List - $II$
($P$) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma=28$, then the system has	($1$) a unique solution
($Q$) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma \neq 28$, then the system has	($2$) no solution
($R$) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma \neq 28$, then the system has	($3$) infinitely many solutions
($S$) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma=28$, then the system has	($4$) $x=11, y=-2$ and $z=0$ as a solution
	($5$) $x=-15, y=4$ and $z=0$ as a solution

Then the system has