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"):}`

If u_(n)=int_(0)^((pi)/(4))tan^(n)xdx , show that, u_(n)+u_(n-2)=(1)/(n-1)(ngt1) , hence find u_(5)

If I_(n)=int_(0)^((pi)/(4)) tan^(n)x dx , then the value of lim_(n to oo) n(I_(n)+I_(n-2)) 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 sin^(n)x dx , show that, I_(n)=-(1)/(n) sin^(n-1)x cosx+(n-1)/(n).I_(n-2) . Hence, evaluate, int sin^(6)x dx .

If I_n = int tan^n x dx , then prove that I_n = (tan^(n-1) x)/(n - 1) - I_(n - 2) . Hence find int tan^7 x dx

Let I_n=int_0^1x^ntan^(-1)xdx." Then prove that "(n+1)I_(n)+(n-1)I_(n-2)=pi/2-1/n"

IfI_n=int_0^1x^n(tan^(-1)x)dx ,t h e np rov et h a t (n+1)I_n+(n-1)I_(n-2)=-1/n+pi/2

If (x^(n^2))^n = (x^(2^n))^2 , then show that root(n+1)(n^3)=2 .

Let I_(n)=int_(0)^(1)x^(n)sqrt(1-x^(2))dx. Then lim_(nrarroo)(I_(n))/(I_(n-2))=