kill(all)$

van=facebook.com/LearningCalculus$

f(x,k):=sin(x)^k;

F21(a,b,c,x):=sum(a*b/c*x^n/n!,n,0,inf)$

F21(1,1,1,x)=%e^x;

integrate(''(f(x,k)),x) =
    -sqrt(2)*cos(x)*sin(x)^(k+1)*
     (sin(x)^2)^(-k/2)*
   ''(F21(1/2,(1-k)/2,3/2,cos(x)^2));

G(x,k):=(-sqrt(2))*cos(x)*sin(x)^(k+1)*
        (sin(x)^2)^((-k)/2)*((1-k)*%e^(cos(x)^2))/6+C;