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

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)=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)^(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)x dx , where n ge 2 , then : I_(n-2)+I_(n)=

I_(n)=int_(0)^((pi)/(4))tan^(n)xdx, then the value of n(l_(n-1)+I_(n+1)) is

If I_(n)=int(ln x)^(n)dx then I_(n)+nI_(n-1)

If I_(n)=int_(1)^(e)(log x)^(n) d x, then I_(n)+nI_(n-1) equal to