# Ex 7.11, 12 - Chapter 7 Class 12 Integrals

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Ex 7.11, 12 By using the properties of definite integrals, evaluate the integrals: _0^ ( )/(1 + sin ) Let I= _0^ /(1+ sin ) I= _0^ ( )/(1+ sin ) Adding (1) and (2) i.e. (1) + (2) I+I= _0^ ( )/(1+ sin ) + _0^ ( )/(1+ sin ) 2I= _0^ ( + )/(1+ sin ) 2I= _0^ ( )/(1+ sin ) I= /2 _0^ ( 1)/(1+ sin ) Solving I(Method 1): I= /2 _0^ ( 1)/(1 + sin ) Multiplying and dividing by cos , I= /2 _0^ ( 1/cos )/( (1+ sin )/cos ) I= /2 _0^ sec /( 1/ cos + ( sin )/cos ) I= /2 _0^ sec /(sec + tan ) Let sec +tan = Differentiating both sides w.r.t. sec tan +sec^2 = / sec [tan +sec ]= / sec [tan +sec ] = sec [ ] = = /( sec ) Putting the values of dx and (sec x+tan x )=t in(3) I= /2 _0^ sec /sec + tan . I= /2 _0^ sec / . I= /2 _0^ sec / /( sec ) I= /2 _0^ 1/ ^2 . I= /2 _0^ ^( 2) I= /2 [ ^( 2 + 1)/( 2 + 1)] _0^ I= /2 [ ^( 1)/( 1)] _0^ I=( )/2 [1/ ] I=( )/2 [1/sec + tan ]_0^ I=( )/2 [1/sec ( ) + tan ( ) 1/sec (0) + tan (0) ] I=( )/2 [1/( 1 + 0) 1/( 1 + 0)] I=( )/2 [ 1 1] I=( )/2 [ 2] I= Solving I (Method 2): I= /2 _0^ ( 1)/(1+ sin ) Multiplying and dividing by (1 sin ) I= /2 _0^ ( 1)/(1+ sin ) (1 sin )/(1 sin ) . I= /2 _0^ (1 sin )/(1 sin^2 ) I= /2 _0^ (1 sin )/( cos ^2 ) I= /2 _0^ [1/cos^2 sin /( cos ^2 )] I= /2 _0^ [sec^2 sin /(cos . cos )] I= /2 _0^ [sec^2 tan sec ] I= /2 [[tan ]_0^ [sec ]_0^ ] I= /2 [[ ( ) (0)] [ ( ) (0)]] I= /2 [[0 0] [ 1 1]] I= /2 [0 ( 2)] I= /2 [2] =

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.