Integrals Of Some Particular Functions

%5Cint%5E%7Be%5E2%7D_%7Be%5E%7B-1%7D%7D%20%5Cbigg%20%7C%20%5Cfrac%7B%20%5Clog_e%20%5C%2C%20x%20%7D%7B%20x%20%7D%20%5Cbigg%20%7C%20%5C%2C%20dx%20%5C%2C%20%3D%20%5Cint%5E%7B1%7D_%7Be%5E%7B-1%7D%7D%20%5Cbigg%20%7C%20%5Cfrac%7B%20%5Clog_e%20%5C%2C%20x%20%7D%7B%20x%20%7D%20%5Cbigg%20%7C%20%5C%2C%20dx%20%5C%2C%20-%20%5Cint%5E%7Be%5E2%7D_%7B%7B1%7D%7D%20%5Cbigg%20%7C%20%5Cfrac%7B%20log_e%20%5C%2C%20x%20%7D%7B%20x%20%7D%20%5Cbigg%20%7C%20%5C%2C%20dx%20%5C%2C

%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5CBigg%20%5B

since , 1 is turning point for

%7C%20%5Cfrac%7B%20%5Clog%20_e%20%5C%2C%20x%20%7D%7Bx%7D%7C

for

%2B%20ve

and

%5C%2C%20-%20ve

values

%5CBigg%20%5D

%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%3D%20-%20%5Cint%5E1_%7Be%5E%7B-1%7D%20%7D%5Cfrac%7B%20%5Clog%20_e%20%5C%2C%20x%20%7D%7Bx%7D%20%5C%2C%20dx%20%2B%20%5Cint%5E%7Be%5E2%7D_1%20%7C%20%5Cfrac%7B%20%5Clog%20_e%20%5C%2C%20x%20%7D%7Bx%7D%7C%20%5C%2C%20dx

%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%3D%20-%5Cfrac%7B1%7D%7B2%7D%20%5B%20(%20%5Clog_e%20%5C%2C%20x%20)%5E2%20%5D%5E1_%7Be%5E%7B-1%7D%7D%20%2B%20%5Cfrac%7B%201%20%7D%7B2%7D%20%5B%20(%20%5Clog%20_%20e%20%5C%2C%20x%20)%5E2%5D%5E%7Be%5E2%20%7D%20_%201

%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%3D%20-%20%5Cfrac%7B1%7D%7B2%7D%20%5Cbig%20%5C%7B%200%20-%20(%20-%201%20)%5E2%20%5Cbig%20%5C%7D%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20(%202%5E2%20-%200%20)%20%3D%20%5Cfrac%7B5%7D%7B2%7D

Given $%5C%2C%20if%20%5C%2C%20I_%20n%20%3D%20%5Cint%5E%7B%5Cpi%7D_%7B%20-%5Cpi%7D%20%5Cfrac%20%7B%20sin%20%5C%2C%20n%20%5C%2C%20x%20%7D%7B%20(%201%20%2B%20%5Cpi%20%5Ex%20)%20%5C%2C%20sin%20%5C%2C%20x%20%7D%20dx%20%2C...............(1)$
Using $%5Cint%5Eb_a%20%5C%2C%20f%20(%20x%20)%20%5C%2C%20dx%20%3D%20%5Cint%5Eb%20_a%20%5C%2C%20f%20(%20b%20%2B%20a%20-%20x%20)%20%5C%2C%20dx%20%2C%20we%20%5C%2C%20get$
$%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20I_n%20%3D%20%5Cint%5E%7B%5Cpi%7D_%20%7B-%5Cpi%20%7D%20%5Cfrac%20%7B%20%5Cpi%5E%7Bx%20%7D%20%5C%2C%20sin%20%5C%2C%20n%20x%7D%7B%20(%201%20%2B%20%5Cpi%5Ex%20)%20sin%20%5C%2C%20x%20%7D%20%5C%2C%20dx%20................(2)$
On adding Eqs. (i) and (ii), we have
$%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%202I_n%20%3D%20%5Cint%5E%7B%5Cpi%7D_%20%7B-%5Cpi%20%7D%20%5Cfrac%20%7B%20sin%20%5C%2C%20n%20x%7D%7B%20sin%20%5C%2C%20x%20%7D%20%5C%2C%20dx%20%3D%202%20%5Cint%5E%7B%5Cpi%7D_%200%20%5Cfrac%20%7B%20sin%20%5C%2C%20n%20x%7D%7B%20sin%20%5C%2C%20x%20%7D%20%5C%2C%20dx$
$%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5B%20%5Cbecause%20f%20(%20x%20)%20%3D%20%5Cfrac%20%7B%20sin%20%5C%2C%20n%20x%7D%7B%20sin%20%5C%2C%20x%20%7D%20%5C%2C%20is%20%5C%2C%20an%20%5C%2C%20even%20%5C%2C%20function%20%5D$
$%5CRightarrow%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C$ $I_n%20%3D%20%5Cint%5E%5Cpi%20_%200%20%5Cfrac%20%7B%20sin%20%5C%2C%20nx%20%7D%7B%20simn%20%5C%2C%20x%20%7D%20%5C%2C%20dx$
$Now%20%2C%20I_%7B%20n%20%2B%202%20%7D%20-%20I_n%20%3D%20%5Cint%5E%7B%5Cpi%7D_0%20%5Cfrac%20%7B%20sin%20(%20n%20-%202%20)%20%5C%2C%20x%20-%20sin%20%5C%2C%20nx%20%7D%7B%20sin%20%5C%2C%20x%20%7D%20%5C%2C%20dx$
$%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%3D%20%5Cint%5E%5Cpi_0%20%5Cfrac%20%7B%202%20%5C%2C%20cos%20(%20n%20%2B%201%20)%20%5C%2C%20x%20.%20sin%20%5C%2C%20x%20%7D%7B%20sin%20%5C%2C%20x%20%7D%20%5C%2C%20dx$
$%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%3D%202%20%5Cint%5E%5Cpi_0%20cos%20(%20n%20%2B%201%20)%20%5C%2C%20x%20%5C%2C%20dx%20%3D%202%20%5Cbigg%20%5B%20%5Cfrac%20%7B%20sin%20(%20n%20%2B%201%20)%20%5C%2C%20x%20%7D%7B%20(%20n%20%2B%201%20)%20%7D%20%5Cbigg%20%5D%5E%7B%5Cpi%7D_0%20%3D%200$
$%5Ctherefore%20%5C%2C%20I_%7B%20n%20%2B%202%20%7D%20%3D%20I_n%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20...................(3)$
$Since%20%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20I_n%20%3D%20%5Cint%5E%7B%5Cpi%7D_0%20%5Cfrac%20%7Bsin%20%5C%2C%20nx%20%7D%7B%20sin%20%5C%2C%20x%20%7D%20%5C%2C%20dx$
$%5CRightarrow$ $%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20I_1%20%3D%20%5Cpi%20%5C%2C%20and%20%5C%2C%20I_2%20%3D%200$
From E (iii) $I_1%20%3D%20I_3%20%3D%20I_5%20%3D%20.................%3D%20%5Cpi$
and $%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C$ $I_2%20%3D%20I_4%20%3D%20I_6%20%3D%20.................%3D%200$
$%5CRightarrow$ $%5Cdisplaystyle%20%5Csum%5E%7B10%7D%20I_%7B2m%20%2B%201%20%7D%20%3D%2010%20%5Cpi%20%5C%2C%20and%20%5C%2C%20%5Cdisplaystyle%20%5Csum%5E%7B10%7D%20I_%7B2m%20%7D%20%3D%200$
$%5Ctherefore$ Correct options are (A), (B), (C).

High-Yield Trend

Chapter Questions
12 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

About Integrals Of Some Particular Functions - JEE-ADVANCED

Frequently Asked Questions

Why focus on Integrals Of Some Particular Functions PYQs?

How to best use this analysis?

Integrals Of Some Particular Functions

High-Yield Trend

Chapter Questions 12 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

About Integrals Of Some Particular Functions - JEE-ADVANCED

Frequently Asked Questions

Why focus on Integrals Of Some Particular Functions PYQs?

How to best use this analysis?

Chapter Questions
12 MCQs