If `I_(m,n) = int cos^m x cdot cos nx cdot dx` show that `(m + n) I_(m,n) = cos^m x cdot sin nx + m I_(m - 1, n-1)`

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)=int_(1)^(e)(logx)^(m)dx , show that, I_(m)+mI_(m-1)lt3 .

If I_(n)=int cos^(n)x dx , prove that, I_(n)=(1)/(n)cos^(n-1)x sin x+(n-1)/(n).I_(n-2) .Hence, evaluate, int cos^(5)x dx .

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(m,n)=int_0^1 t^m(1+t)^n.dt , then the expression for I(m,n) in terms of I(m+1,n-1) is:

If I_(m)=int_(1)^(e)(log_(e)x)^(m)dx , then prove that, I_(m)=e-mI_(m-1)

If I_(m)=int_(1)^(e )(log_(e)x)^(m)dx , then the value of (I_(m)+mI_(m-1)) is -

If I_m=int_1^x(logx)^mdx satisfies the relation (I_m)=k-l I_(m-1) then

Given I_m=int_1^e(logx)^mdx ,then prove that (I_m)/(1-m)+m I_(m-2)=e

IfI_(m , n)=intcos^m xsinn xdx , then 7I_(4,3)-4I_(3,2)" is equal to" (a) constant (b) -cos^2x+C (c) -cos^4xcos3x+C (d) cos7x-cos4x+C