Evaluate `underset(n=1)overset(89)Sigma(1)/(1+(tann^(@))^(2)`

Evaluate underset(1)overset(9) ((2t^(2)+t^(1/2)sqrt(t)-1)/(t^2)dt)

Let f(n), g (n) be polynomial in n and let Delta_(r)=|{:(r^(2).C_(r),n(n+1)2^(n-2),f(n)),(6r(n-r+1),n(n+1)(n+2),g(n)),(4(5^(r-1)),5^(n)-1,h(n)):}| , then evaluate underset(r=1)overset(n)sum Delta_(r) . (use (1+x)^(n)=underset(r=0)overset(n)sum(C_(r)x^(r)) )

The value of cot[underset(n=1)overset(100)Sigmacot^(-1)(1+underset(k=1)overset(n)Sigma2k)] is

underset(r=1)overset(n)sum ( r )/(r^(4)+r^(2)+1) is equal to

The sum, underset(n-1)overset(10)(Sigma)(n(2n+1)(2n-1))/5 is equal to.........

The limit of the series underset(r=1)overset(n)sum (r)/(1+r^(2)+r^(4)) as n approaches infinity, is

Evaluate the following definite integrals underset(0)overset(1)int(dx)/(sqrt(1-x^(2)))