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,n)=int_0^1x^(m-1)(1-x)^(n-1)dx , then

If L(m,n)=int_(0)^(1)t^(m)(1+t)^(n),dt , then prove that L(m,n)=(2^(n))/(m+1)-n/(m+1)L(m+1,n-1)

If I_(m,n)= int_(0)^(1) x^(m) (ln x)^(n) dx then I_(m,n) is also equal to

If I_(n)=int cos^(n)x dx . Prove that I_(n)=(1)/(n)(cos^(n-1)x sinx)+((n-1)/(n))I_(n-2) .

If y=(x-a)^(m)(x-b)^(n) , prove that (dy)/(dx)=(x-a)^(m-1)(x-b)^(n-1)[(m+n)x-(an+bm) ].

If I_(n)=int(sinx+cosx)^(n) dx, snd I_(n)=1/n(sinx+cosx)^(n-1)(sinx-cosx)+(2k)/(n) I_(n-2) then k=

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_(m"," n)=int cos^(m)x*cos nx dx , show that (m+n)I_(m","n)=cos^(m)x*sin nx+m I_((m-1","n-1))

If I_n=intxcos^nxdx,prove that: I_n=1/n cos^(n-1)x(sinx)+(n-1)/(n)I_(n-2)

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