restart;
f:=1/4+4*t^2*(1-t);
g:=1/4+sin(Pi*t);
df:=diff(f,t);
dg:=diff(g,t);
A:=int(dg*f,t=0..1);
Int(dg*f,t=0..1);
evalf(A);
P1:=plot([f,g,t=0..1],view=[0..1.5,0..1.5]):
P2:=plot([[3/4,5/4]],style=point):
with(plots):
display(P1,P2);
x0:=subs(t=1/2,f);
y0:=subs(t=1/2,g);
simplify(y0);
m:=subs(t=1/2,dg/df);
simplify(m);
series(exp(x^2),x=0,6);