Show that :
`x^(-n ) = (1/x^n )`

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

If x + y = 1, show that D ^ (n) (x ^ (n) y ^ (n)) = n! [Y ^ (n) - (nC_ (1)) ^ (2) y ^ (n -1) x + (nC_ (2)) ^ (2) y ^ (n-2) x ^ (2) + ...... + (- 1) ^ (n) x ^ (n)]

Let a_(0)/(n+1) +a_(1)/n + a_(2)/(n-1) + ….. +(a_(n)-1)/2 +a_(n)=0 . Show that there exists at least one real x between 0 and 1 such that a_(0)x^(n) + a_(1)x^(n-1) + …..+ a_(n)=0

Show that (1+x)^(n) - nx-1 is divisible by x^(2) for n gt N

Show that int sin^(n)xdx=(-sin^(n-1)x*cos x)/(n)+(n-1)/(n)int sin^(n-2)ndx also find int sin^(6)dx

A sequence x_(1),x_(2),x_(3),.... is defined by letting x_(1)=2 and x_(k)=(x_(k-1))/(k) for all natural numbers k,k>=2 Show that x_(n)=(2)/(n!) for all n in N.

Show that: (x)+(x+(1)/(n))+(x+(2)/(n))+...+(x+(n-1)/(n))=nx+(n-1)/(2)

Show that 1+2x + 3x^2 +….+ nx^(n-1) = (1-(n+1)x^(n) + nx^(n+1))/((1-x)^2) for all n in N .

Using the principle of mathematical induction to show that tan^(-1)(n+1)x-tan^(-1)x , forall x in N .

If y=x^(2)e^(x) ,show that y_(n)=(1)/(2)n(n-1)y_(2)-n(n-2)y_(1)+(1)/(2)(n-1)(n-2)}