If `I_(n)=int_(0)^(pi/2) sin^(n)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"):}`

Let I_(n)=int_(0)^(pi//2) sin^(n)x dx, nin N . Then

If I_n=int_0^(pi//4)tan^("n")x dx , prove that I_n+I_(n-2)=1/(n-1)dot

If l_(n)=int_(0)^(pi//4) tan^(n)x dx, n in N "then" I_(n+2)+I_(n) equals

If I_(n)=int_(0)^(pi) e^(x)sin^(n)x " dx then " (I_(3))/(I_(1)) is equal to

I_n = int_0^(pi/4) tan^n x dx , then the value of n(l_(n-1) + I_(n+1)) is

IfI_(m , n)=int_0^(pi/2)sin^m xcos^n xdx , Then show that I_(m , n)=(m-1)/(m+n)I_(m-2,n)(m ,n in N) Hence, prove that I_(m , n)=f(x)={((n-1)(n-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))pi/4 when both m and n are even ((m-1)(m-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))}

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_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_0^ooe^(-x)x^(n-1)log_exdx , then prove that I_(n+2)-(2n+1)I_(n+1)+n^2I_n=0

Let I_(n)=int_(0)^(pi//2)(sinx+cosx)^(n)dx(nge2) . Then the value of n. I_(n)-2(n-1)I_(n-1) is