GATE CSE
First time here? Checkout the FAQ!
x
+9 votes
1.1k views

If for non-zero $x, \: af(x) + bf(\frac{1}{x}) = \frac{1}{x} - 25$ where a $a \neq b \text{ then } \int_1^2 f(x)dx$ is

 

  1. $\frac{1}{a^2 - b^2} \begin{bmatrix} a(\ln 2 - 25) + \frac{47b}{2} \end{bmatrix}$
  2. $\frac{1}{a^2 - b^2} \begin{bmatrix} a(2\ln 2 - 25) - \frac{47b}{2} \end{bmatrix}$
  3. $\frac{1}{a^2 - b^2} \begin{bmatrix} a(2\ln 2 - 25) + \frac{47b}{2} \end{bmatrix}$
  4. $\frac{1}{a^2 - b^2} \begin{bmatrix} a(\ln 2 - 25) - \frac{47b}{2} \end{bmatrix}$
asked in Calculus by Veteran (92.5k points) 964 2325 3111
edited by | 1.1k views

In the equation, substitute (1/x) for x, then you get another equation. Now solve the given equation and the obtained equation. You will get f(x). And then integrate. Simple calculations.

Answer: A 

1 Answer

+25 votes
Best answer

$af\left ( x \right )+bf\left ( \frac{1}{x} \right )=\frac{1}{x} -25$ --- $\left ( 1 \right )$


Integrating both sides,

$a\int_{1}^{2}f\left ( x \right )dx+b\int_{1}^{2}f\left ( \frac{1}{x} \right )dx=\left [ \log\left ( x \right )-25x \right ]_{1}^{2}=\log2-25$ --- $\left ( 2 \right )$


Replacing  $x$ by $\frac{1}{x}$ in $\left ( 1 \right )$, we get

$af\left ( \frac{1}{x} \right )+bf\left ( x \right )=x-25$

Integrating both sides, we get

$a\int_{1}^{2}f\left ( \frac{1}{x} \right )dx+b\int_{1}^{2}f\left ( x \right )dx=\left [ \frac{x^{2}}{2}-25x \right ]_{1}^{2}=-\frac{47}{2}$ --- $\left ( 3 \right )$


Eliminate $\int_{1}^{2}f\left ( \frac{1}{x} \right )$ between $\left ( 2 \right )$ and $\left ( 3 \right )$ by multiplying $\left ( 2 \right )$ by $a$ and $\left ( 3 \right )$ by $b$ and subtracting

$\therefore \left ( a^{2}-b^{2} \right )\int_{1}^{2}f\left ( x \right )dx=a\left ( \log2-25 \right )+b\times\frac{47}{2}$

$\therefore \int_{1}^{2}f\left ( x \right )dx=\frac{1}{\left ( a^{2}-b^{2} \right )}\left [ a\left ( \log2-25 \right )+\frac{47b}{2} \right ]$


Answer: A. $\frac{1}{\left ( a^{2}-b^{2} \right )}\left [ a\left ( \log2-25 \right )+\frac{47b}{2} \right ]$

answered by Active (1.5k points) 8 12 20
selected by
Answer:

Related questions

+6 votes
4 answers
2
+4 votes
4 answers
3


Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
Top Users Oct 2017
  1. Arjun

    23324 Points

  2. Bikram

    17048 Points

  3. Habibkhan

    7808 Points

  4. srestha

    6222 Points

  5. Debashish Deka

    5430 Points

  6. jothee

    4958 Points

  7. Sachin Mittal 1

    4772 Points

  8. joshi_nitish

    4286 Points

  9. sushmita

    3964 Points

  10. Rishi yadav

    3794 Points


Recent Badges

Popular Question iarnav
Notable Question makhdoom ghaya
Popular Question LavTheRawkstar
Avid Voter atul_21
Popular Question hem chandra joshi
100 Club nikhil_cs
Notable Question Sukannya
Notable Question Sourabh Kumar
Notable Question shikharV
Nice Comment Sachin Mittal 1
27,287 questions
35,134 answers
83,911 comments
33,223 users