CBSE Class 12-science Answered

Integration of x2 divide by x4+x2-2

Asked by mpsaran03 | 16 Mar, 2019, 08:13: PM

Expert Answer

begin mathsize 12px style I space equals space integral fraction numerator x squared d x over denominator x to the power of 4 plus x squared minus 2 end fraction space open parentheses x to the power of 4 plus x squared minus 2 close parentheses space equals space open parentheses x squared plus 1 half close parentheses squared minus open parentheses 3 over 2 close parentheses squared f o r space c o n v e n i e n c e space l e t space u s space p u t space space a space equals 1 half space space a n d space space space b space equals space 3 over 2 h e n c e space I space equals space integral fraction numerator x squared d x over denominator open parentheses x squared plus a close parentheses squared minus b squared end fraction space equals 1 half space integral space space x space fraction numerator space 2 x d x over denominator open parentheses x squared plus a close parentheses squared minus b squared end fraction space equals space 1 half integral u space d v w h e r e space u space equals space x space comma space d v space equals fraction numerator space 2 x d x over denominator open parentheses x squared plus a close parentheses squared minus b squared end fraction h e n c e space d u space equals space d x space space comma space v space equals space integral fraction numerator space 2 x d x over denominator open parentheses x squared plus a close parentheses squared minus b squared end fraction space equals space fraction numerator 1 over denominator 2 b end fraction space log open parentheses fraction numerator x squared plus a minus b over denominator x squared plus a plus b end fraction close parentheses h e n c e space I space equals space fraction numerator x over denominator 4 b end fraction space log open parentheses fraction numerator x squared plus a minus b over denominator x squared plus a plus b end fraction close parentheses space minus space fraction numerator 1 over denominator 4 b end fraction integral log open parentheses fraction numerator x squared plus a minus b over denominator x squared plus a plus b end fraction close parentheses d x space space space space I space equals space fraction numerator x over denominator 4 b end fraction space log open parentheses fraction numerator x squared plus a minus b over denominator x squared plus a plus b end fraction close parentheses space minus space fraction numerator 1 over denominator 4 b end fraction integral space log open parentheses x squared plus a minus b close parentheses d x space space plus space space fraction numerator 1 over denominator 4 b end fraction integral space log open parentheses x squared plus a plus b close parentheses d x space space......... left parenthesis 1 right parenthesis end style

Let us work out a integration I₁ =

begin mathsize 12px style integral log left parenthesis space x squared plus k right parenthesis space d x end style

begin mathsize 12px style I subscript 1 space equals space integral log left parenthesis x squared plus k right parenthesis space d x space space equals space integral u space d v w h e r e space u space equals space log open parentheses x squared plus k close parentheses space space space a n d space space d u space equals space fraction numerator 2 x space d x over denominator open parentheses x squared plus k close parentheses end fraction d v space equals space d x space comma space space space v space space equals space x I subscript 1 space equals space x space log open parentheses x squared plus k close parentheses space minus integral fraction numerator 2 space x squared space d x over denominator open parentheses x squared plus k close parentheses end fraction space space equals space x space log open parentheses x squared plus k close parentheses space minus 2 space I subscript 2 space space................... left parenthesis 3 right parenthesis w h e r e space I subscript 2 space equals space integral fraction numerator space x squared space d x over denominator open parentheses x squared plus k close parentheses end fraction space equals space integral fraction numerator space x squared space plus k space minus space k space d x over denominator open parentheses x squared plus k close parentheses end fraction space equals space integral space d x space minus space k integral fraction numerator d x over denominator open parentheses x squared plus k close parentheses end fraction I subscript 2 space equals space x space minus space fraction numerator k over denominator square root of k end fraction space tan to the power of negative 1 end exponent open parentheses fraction numerator x over denominator square root of k end fraction close parentheses space equals space x space minus space space square root of k space tan to the power of negative 1 end exponent open parentheses fraction numerator x over denominator square root of k end fraction close parentheses h e n c e space w e space w r i t e space I subscript 1 space a s space comma space space space space I subscript 1 space equals space x space log open parentheses x squared plus k close parentheses space minus 2 space x space plus space 2 space space square root of k space tan to the power of negative 1 end exponent open parentheses fraction numerator x over denominator square root of k end fraction close parentheses space space........... left parenthesis 4 right parenthesis end style

let use this integration I₁ in eqn.(1) and simplify the resulting expression

begin mathsize 12px style I space equals space fraction numerator x over denominator 4 b end fraction log fraction numerator x squared plus a minus b over denominator x squared plus a plus b end fraction space minus space fraction numerator 1 over denominator 4 b end fraction open curly brackets x space log open parentheses x squared plus a minus b close parentheses space minus space 2 x space plus space 2 square root of a minus b end root space tan to the power of negative 1 end exponent open parentheses fraction numerator x over denominator square root of a minus b end root end fraction close parentheses space minus space x space log open parentheses x squared plus a plus b close parentheses space plus 2 x space minus space 2 square root of a plus b end root space tan to the power of negative 1 end exponent open parentheses fraction numerator x over denominator square root of a plus b end root end fraction close parentheses close curly brackets I space equals space 2 open curly brackets square root of a plus b end root space tan to the power of negative 1 end exponent open parentheses fraction numerator x over denominator square root of a plus b end root end fraction close parentheses space minus square root of a minus b end root space tan to the power of negative 1 end exponent open parentheses fraction numerator x over denominator square root of a minus b end root end fraction close parentheses space space space close curly brackets B y space s u b s t i t u t i n g space v a l u e s space f o r space a space a n d space b comma space a b o v e space e x p r e s s i o n space b e c o m e s I space equals space 2 open curly brackets square root of 2 space tan to the power of negative 1 end exponent open parentheses fraction numerator x over denominator square root of 2 end fraction close parentheses space minus square root of negative 1 end root space tan to the power of negative 1 end exponent open parentheses fraction numerator x over denominator square root of negative 1 end root end fraction close parentheses space space space close curly brackets l e t space u s e space t h e space d e f i n i t i o n space square root of negative 1 end root space equals space i space comma space i m a g i n a r y space n u m b e r space u s e d space c o m p l e x space n u m b e r s I space equals space 2 open curly brackets square root of 2 space tan to the power of negative 1 end exponent open parentheses fraction numerator x over denominator square root of 2 end fraction close parentheses space minus i space space tan to the power of negative 1 end exponent open parentheses x over i close parentheses space space space close curly brackets end style

Answered by Thiyagarajan K | 17 Mar, 2019, 03:49: PM

CBSE Class 12-science Answered

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CBSE Class 12-science Answered

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