Prove that `int_(0)^(1)(dx)/(1+x^(n))gt1-(1)/(n)"for n"inN`

int_(0)^(1)x(1-x)^(n)dx

Prove that: int_(-1)^(1)x|x|dx=0

int_(0)^(oo)(dx)/([x+sqrt(1+x^(2))]^(n)

Prove that lim_(xrarr0) ((1+x)^(n) - 1)/(x) = n .

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

Prove that, int_(0)^((pi)/(2))cos^(n)x cos nx dx=(pi)/(2^(n+1)) .

int(dx)/(xsqrt(x^(n)+1))

Prove that: I_n=int_0^oox^(2n+1) e^ (-x^2) dx=(n !)/2,n in N .

Prove that .^(n)C_(0) + (.^(n)C_(1))/(2) + (.^(n)C_(2))/(3) + "……" +(. ^(n)C_(n))/(n+1) = (2^(n+1)-1)/(n+1) .

Prove that sum_(k=0)^(n) (-1)^(k).""^(3n)C_(k) = (-1)^(n). ""^(3n-1)C_(n)