If `I_(m)=int (sin x+cos x)^(m)dx`, then show that `m l_(m)=(sin x+ cos x)^(m-1)*(sin x- cos x)+2(m-1) I_(m-2)`

int(sin^(m)x)/(cos^(m+2)x)dx=

If I_(m;n)=int_(0)^((pi)/(2))sin^(m)x cos^(n)xdx then show that I_(m;n)=(m-1)/(m+n)I_(m-2;n) and find I_(m;n) in terms of different combinations of m and n.

sin^(m)x.cos^(n)x

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(m)=int_(0)^( pi)ln(1-2m cos x+m^(2))dx then I(1)=

If l_(n)=int e^(mx)cos^(n)xdx then prove that (m^(2)+n^(2))I_(n)=e^(mx)*(m cos x+n sin x)cos^(n-1)x+n(n-1)l_(n)

Given I_(m)=int_(1)^(e)(log x)^(m)dx, then prove that (I_(m))/(1-m)+mI_(m-2)=e

(i) int (sin x)/(1+ cos x) dx. (ii) int (sin x)/(1+ cos x)^2 dx.

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