If `I_n=int_0^1(dx)/((1+x^2)^n); n in N ,` then prove that `2nI_(n+1)=2^(-n)+(2n-1)I_n`

If I_n=intdx/(x^2+a^2)^n,ninN , then show that: I_(n+1)=1/(2na^2)x/((x^2+a^2)^n)+(2n-1)/(2n). 1/a^2I_n

If I_(n)=int(x^(n)dx)/(sqrt(x^(2)+a)) then prove that I_(n)+(n-1)/(n)al_(n-2)=(1)/(n)x^(n-1)*sqrt(x^(2)+a)

If I_n=int_0^ooe^(-x)x^(n-1)log_exdx , then prove that I_(n+2)-(2n+1)I_(n+1)+n^2I_n=0

If I_(n)=int_(0)^(1)x^(n)(tan^(-1)x)dx, then prove that(n+1)I_(n)+(n-1)I_(n-2)=-(1)/(n)+(pi)/(2)

If I_(n)=int(x^(n))/(1+x^(2))dx, where n in N , then : I_(n+2)+I_(n)=

For n in N, prove that (n+1)[n!n+(n-1)!(2n-1)+(n-2)!(n-1)]=(n+2)!

If I_(n)=int_(0)^(pi/2) sin^(x)x dx , then show that I_(n)=((n-1)n)I_(n-2) . Hence prove that I_(n)={(((n-1)/n)((n-3)/(n-2))((n-5)/(n-4))………(1/2)(pi)/2,"if",n"is even"),(((n-1)/n)((n-3)/(n-2))((n-5)/(n-4))………(2/3)1,"if",n"is odd"):}

If I_(n)=int_(0)^( pi/4)tan^(n)xdx, prove that I_(n)+I_(n-2)=(1)/(n+1)

If u_(n)=int_(0)^((pi)/(2))theta sin^(n)theta d theta and n>=1, then prove that u_(n)=((n-1)/(n))u_(n-2)+(1)/(n^(2))

Prove that ((2n+1)!)/(n!)=2^(n){1.3.5(2n-1)(2n+1)}