" (vi) "x^(n)(log x)^(2)

"int x^(n)(log x)^(2)dx

(d^(n))/(dx^(n))(log x)=(a)((n-1)!)/(x^(n))(b)(n!)/(x^(n))(c)((n-2)!)/(x^(n))(d)(-1)^(n-1)((n-1)!)/(x^(n))

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

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

Compute the integrals: int_(oo)f(x^(n)+x^(-n))log x(dx)/(1+x^(2))0

If n is a multiple of 3, then coefficient of x^(n) in log(1+x+x^(2)) is

Let I_(n) =int _(1) ^(e^(2))(ln x)^(n) dx (x ^(2)), then the value of 3I_(n)+nI_(n-1) equals to:

(i) (log x. sin[1+(log x)^(2)])/(x) dx (ii) int(dx)/(x(1+log)^(n))