Given `I_m=int_1^e(logx)^mdx ,t h e np rov et h a t(I_m)/(1-m)+m I_(m-2)=e`

If I_(m)=int_(1)^(x) (log x)^(m)dx satisfies the relation I_m = k-lI_(m-1) then,

Let I_m=int_0^pi (1-cosmx)/(1-cosx)dx . Show that I_m=mpi .

If I_(m,n)= int(sinx)^(m)(cosx)^(n) dx then prove that I_(m,n) = ((sinx)^(m+1)(cosx)^(n-1))/(m+n) +(n-1)/(m+n). I_(m,n-2)

If I_(m)=int_(1)^(e)(ln x)^(m)dx,m in N, then I_(10)+10I_(9) is equal to (A) e^(10)(B)(e^(10))/(10)(C)e(D)e-1

If I_(n)=int_(1)^(e)(ln x)^(n)dx(n is a natural number) then I_(2020)+nI_(m)=e, where n,m in N .The value of m+n is equal to

IfI_(m , n)=int_0^(pi/2)sin^m xcos^n xdx , Then show that I_(m , n)=(m-1)/(m+n)I_m-2n(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/4w h e nbot hma n dna r ee v e n((m-1)(m-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))

if I_(m,n)=int(x^(m))/((log x)^(n))dx, then (m+1)I_(m,n)-nI_(m,n+1) is