A function f(x) satisfies f ( x ) = f ( c x ) for some real numbe… - Vidyadip

Question

A function $f(x)$ satisfies $f\left( x \right) = f\left( {\frac{c}{x}} \right)$ for some real number $c\left( {c > 1} \right)$ and $\forall\, x > 0$. If $\int\limits_1^{\sqrt c } {\frac{{f\left( x \right)}}{x}} dx = 3$ , then the value of $\int\limits_1^c {\frac{{f\left( x \right)}}{x}} dx$ is

✓

Answer

d
Put $u=\frac{c}{x} \Rightarrow d u=\frac{-c}{x^{2}} d x$

$\int_1^{\sqrt e } {\frac{{f(x)}}{x}} dx = \int_e^{\sqrt e } {\frac{{uf(u)}}{c}} \left( { - \frac{{{x^2}}}{c}} \right)du = \int_{\sqrt e }^e f (u)du$

$\int_1^e {\frac{{f(x)}}{x}} dx = \int_1^{\sqrt e } {\frac{{f(x)}}{x}} dx + \int_{\sqrt e }^e {\frac{{f(u)}}{u}} dx = 6$

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

A fair die is tossed repeatedly until a six is obtained. Let $\mathrm{X}$ denote the number of tosses required and let $\mathrm{a}=\mathrm{P}(\mathrm{X}=3), \mathrm{b}=\mathrm{P}(\mathrm{X} \geq 3)$ and $\mathrm{c}=$ $\mathrm{P}(\mathrm{X} \geq 6 \mid \mathrm{X}>3)$. Then $\frac{\mathrm{b}+\mathrm{c}}{\mathrm{a}}$ is equal to

Let $y=y(x)$ be solution of the following differential equation $e^{y} \frac{d y}{d x}-2 e^{y} \sin x+\sin x \cos ^{2} x=0, y$ $\left(\frac{\pi}{2}\right)=0$. If $y(0)=\log _{e}\left(\alpha+\beta e^{-2}\right)$, then $4(\alpha+\beta)$ is equal to $....$

If ${\cos ^{ - 1}}\frac{3}{5} - {\sin ^{ - 1}}\frac{4}{5} = {\cos ^{ - 1}}x,$ then $ x=$

The value of ${\log _3}\,4{\log _4}\,5{\log _5}\,6{\log _6}\,7{\log _7}\,8{\log _8}\,9$ is

If $\sum\limits_{K = 1}^{12} {12K{.^{12}}{C_K}{.^{11}}{C_{K - 1}}} $ is equal to $\frac{{12 \times 21 \times 19 \times 17 \times ........ \times 3}}{{11!}} \times {2^{12}} \times p$ then $p$ is

If $\int \frac{d x}{\left(x^{2}+x+1\right)^{2}}=a \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+b\left(\frac{2 x+1}{x^{2}+x+1}\right)+C$ $x>0$ where $C$ is the constant of integration, then the value of $9(\sqrt{3} \mathrm{a}+\mathrm{b})$ is equal to ... .

The number of rectangles that can be obtained by joining four of the twelve vertices of a $12$ -sided regular polygon is

The number of ways in which a committee of $6$ members can be formed from $8 $ gentlemen and $4$ ladies so that the committee contains at least $3$ ladies is

Let $\alpha=\sum_{\mathrm{r}=0}^{\mathrm{n}}\left(4 \mathrm{r}^2+2 \mathrm{r}+1\right)^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}$ and $\beta=\left(\sum_{\mathrm{r}=0}^{\mathrm{n}} \frac{{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}}{\mathrm{r}+1}\right)+\frac{1}{\mathrm{n}+1}$. If $140<\frac{2 \alpha}{\beta}<281$ then the value of $n$ is...............

If the remainder when $x$ is divided by $4$ is $3 ,$ then the remainder when $(2020+ x )^{2022}$ is divided by $8$ is ....... .