The following question consist of two statement, one labelled as the "Assertion (A)' and the other as 'Reason (R)'. You are are to examine these two statement carefully and select the answer.
Assertion (A) : `int_(1)^(e ) ln^(2) xdx = e - 2`
Reason (R ) : `I_(n) = int_(1)^(e) ln^(n) xdx = e- nI_(n-1)`

Options

A. Both A and R are individually true and R is the correct explanation of A
B. Both A and R are individually true and R is not the correct explanation of A
C. A is true but R is false
D. A is false but R is true.

Answer

A

Explanation

Assertion (A) `: int_(1)^€ ln^(2)x. dx = int_(1)^(e ) ln^(2) x.1 dx`
`intln^(2)x. 1dx = x.ln^(2)(x) - int2.ln x.dx`
`:. Int_(1)^(e ) ln^(2)x.dx = [x.ln^(2)x-2xlnx + 2x]_(1)^(e )`.
`= (e. ln^(2) e - 2e. Lne + 2e) - (ln^(2) 1-2ln1+2)`
`= e - 2e + 2e - 0 + 0 - 2 ( :' ln e = 1 "and" ln 1 = 0)`
`= e - 2`
So, A is true.
Reason (R ) : `I_(n) = int_(1)^(e) ln^(n)x.dx`
`:. intln^(n) x.dx = x xx ln^(n) x - n int ln^(n--1)x.dx`
`= x. ln^(n) x - n xx I_(n-1)`
So, `int_(1)^(e) ln^(n) x.dx = [x. ln^(x)x - n xx I_(n-1)]_(1)^(e)`
`= e. ln^(n) e - ln^(n) 1 - n xx I_(n-1)`
`= e- n xx I_(n-1) ( :' ln e = 1 "and" ln 1 = 0)`
So, R is true and R is correct explanation of A.