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

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

Chapter Questions
12 MCQs