Integration x . log(x + 1)  dx explain in great detail and give full solution

Asked by haroonrashidgkp | 18th Oct, 2018, 04:30: PM

Expert Answer:

u s e space L I A T E
left parenthesis log a r i t h m comma i n v e r s e comma a l g e b r a i c comma t r i g n o m e t r y comma e x p o n e n t i a l right parenthesis
a c c o r d i n g l y space d e c i d e space u space a n d space v.
i n space o u r space c a s e space x. log left parenthesis x plus 1 right parenthesis space s o space b y space L I A T E space t a k e space u equals log left parenthesis x plus 1 right parenthesis space a n d space v equals x.
s o space i n t e g r a t i o n space b y space p a r t s
integral u v space d x equals u integral v d x minus integral left square bracket fraction numerator d u over denominator d x end fraction integral v d x right square bracket d x
integral log left parenthesis x plus 1 right parenthesis x d x equals log left parenthesis x plus 1 right parenthesis integral x d x minus integral left square bracket fraction numerator d log left parenthesis x plus 1 right parenthesis over denominator d x end fraction integral x d x right square bracket d x
equals log left parenthesis x plus 1 right parenthesis space x squared over 2 minus integral fraction numerator 1 over denominator x plus 1 end fraction. x squared over 2 space d x
equals log left parenthesis x plus 1 right parenthesis space x squared over 2 minus 1 half integral fraction numerator x squared over denominator x plus 1 end fraction. space d x
n o w space root index x plus 1 of x squared end root space comma i. e space d i v i d e space x squared space b y space x plus 1
w e space g e t space x squared equals left parenthesis x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis plus 1
equals log left parenthesis x plus 1 right parenthesis space x squared over 2 minus 1 half integral fraction numerator left parenthesis x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis plus 1 over denominator x plus 1 end fraction. space d x
equals log left parenthesis x plus 1 right parenthesis space x squared over 2 minus 1 half integral fraction numerator left parenthesis x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis over denominator x plus 1 end fraction plus fraction numerator 1 over denominator x plus 1 end fraction. space d x
equals log left parenthesis x plus 1 right parenthesis space x squared over 2 minus 1 half integral x minus 1 plus fraction numerator 1 over denominator x plus 1 end fraction. space d x
equals log left parenthesis x plus 1 right parenthesis space x squared over 2 minus 1 half left square bracket x squared over 2 minus x plus log vertical line x plus 1 vertical line right square bracket plus space c
equals log left parenthesis x plus 1 right parenthesis space x squared over 2 minus x squared over 4 plus x over 2 minus 1 half log vertical line x plus 1 vertical line right square bracket plus space c

Answered by Ram Singh | 18th Oct, 2018, 07:50: PM