maple, expansion of definite integral to series

Question

i have trouble with this following integral due to the s upper limit. i have tried series and expand , but i understond the assigment is to use convert(series()).enter image description here

int(sin(x*(y^(1/2)+y^(1/3)), y = 0..x)

i tried series, convert, expand. expected result is the expansion to series of thr integral, which is (3/4)x^(7/3)+(2/5)x^(5/2)+(-1/12)x^5+O(7)

if i swap the upper limit with any number i receive a correct series for the.given limit. this is an exampleenter image description here

Answer 1

Is this the kind of this you're hoping to accomplish?

restart;

G := proc(n::posint, v::name:=':-x') local T,y;
  T:=series(int(sin(v*(y^(1/2)+y^(1/3))), y), v, n);
  T:=expand(convert(eval(T,y=v)-eval(T,y=0),polynom));
  T:=select(t->op(2,indets(t,identical(v)^numeric)[1])<=n,T);
  sort(T,'order'=':-plex'(v),':-ascending');
end proc:

G(5);

    3/4*x^(7/3)+2/3*x^(5/2)-1/12*x^5

G(7);

    3/4*x^(7/3)+2/3*x^(5/2)-1/12*x^5-3/13*x^(31/6)
    -3/14*x^(16/3)-1/15*x^(11/2)

G(8);

    3/4*x^(7/3)+2/3*x^(5/2)-1/12*x^5-3/13*x^(31/6)
    -3/14*x^(16/3)-1/15*x^(11/2)+1/320*x^(23/3)
    +1/68*x^(47/6)+1/36*x^8