" 2."int x^(n)log xdxquad " [Ans: "(x^(n+1))/([n+1]^(2))[(n+1)log x-1]+C]

int_1^(e) x^(n) log x dx

" (1) "int((1+log x)^(n))/(x)dx

Assertion (A) : coefficient x^(n) in log(1+x) is ((-1)^(n-1))/(n) when |x| lt 1 . Reason (R ) : Coefficient of x^(n) in log(1-x) is -1//2 when |x| lt 1 .

Show that (d^n)/(dx^(n) )(x^(n) log x) = n! (log x + 1+(1)/(2) +…+(1)/(n)) AA n in N .

int((1+log x)^(n))/(x)dx x>(1)/(e) n!=-1

Evaluate the integerals. int x ^(n) log x dx on (0,oo) , n is a real nuber and n ne -1.

Prove that D^(n)((log x)/(x))=(-1)^(n)n!x^(-n-1)[log x-1-(1)/(2)-(1)/(3)-......(1)/(n)]

If I_(n)= int (log x)^(n)dx then I_(n)+nI_(n-1)=

If n is not a multiple of 3, then the coefficient of x^n in the expansion of log_e (1+x+x^2) is : (A) 1/n (B) 2/n (C) -1/n (D) -2/n

If I_(n)=int(ln x)^(n)dx then I_(n)+nI_(n-1)