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Chapter 7 Class 12 Integrals
Concept wise

Ex 7.5, 7 - Integrate x / (x2 + 1) (x - 1) - Class 12 NCERT - Integration by partial fraction - Type 5

Ex 7.5, 7 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.5, 7 - Chapter 7 Class 12 Integrals - Part 3
Ex 7.5, 7 - Chapter 7 Class 12 Integrals - Part 4
Ex 7.5, 7 - Chapter 7 Class 12 Integrals - Part 5
Ex 7.5, 7 - Chapter 7 Class 12 Integrals - Part 6

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Transcript

Ex 7.5, 7 /( ^2 + 1)( 1) Let I= 1 /( ^2 + 1)( 1) We can write integrand as /( ^2 + 1)( 1) =( + )/(( ^2 + 1) ) + /(( 1) ) /( ^2 + 1)( 1) =(( + )( 1) + ( ^2 + 1))/( ^2 + 1)( 1) By Cancelling denominator =( + )( 1)+ ( ^2+1) Putting x = 1, in (1) =( + )( 1)+ ( ^2+1) 1=( 1+ )(1 1)+ (1^2+1) 1=( + ) 0+ (1+1) 1=0+2 =1/2 Putting x = 0 , in (1) =( + )( 1)+ ( ^2+1) 0=( 0+ )(0 1)+ (0+1) 0 = ( 1)+ = =1/2 Putting x = 1 1=( ( 1)+ )( 1 1)+ ((1)^2+1) 1=( + )( 2)+ (1+1) 1=2 2 +2 1=2 2 1/2+2 1/2 1=2 1+1 1=2 =( 1)/2 Hence we can write /( ^2 + 1)( 1) =(( 1/2 + 1/2))/(( ^2 + 1) ) + (1/2)/(( 1) ) =( 1 . )/(2 ( ^2 + 1) ) + 1/(2 ( ^2 + 1) )+1/2( 1) Integrating . . . I= 1 /( ^2 + 1)( 1) = 1/2 1 /( ^2 + 1) +1/2 1 /( ^2 + 1) + 1/2 1 /( 1) Solving I1= 1/2 1 /(( ^2 + 1) ) Put ^2+1= Different both sides w.r.t. 2 +0= / = /2 Hence we can write 1/2 1 /(( ^2 + 1) ) = 1/2 1 / /2 = 1/(2 2) 1 / = 1/4 log | |+ 1 = 1/4 log | ^2+1|+ 1 Solving I2=1/2 1 1/( ^2 + 1) =1/2 tan^( 1) + 2 Solving I3=1/2 1 1/( 1) =1/2 log | 1|+ 3 Hence I=I1+I2+I3 = 1/4 log | ^2+1|+ 1+1/2 tan^( 1) + 2+1/2 log | 1|+ 3 = 1/4 log | ^2+1|+1/2 tan^( 1) +1/2 log | 1|+ = 1/4 log | ^2+1|+1/2 tan^( 1) +1/2 log | 1|+ = / | | / | ^ + |+ / ^( ) + C

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.