(∫_0^1.)Tan[x]\[DifferentialD]x

Integrate[Tan[x],{x,0,1.}]

(∫_0^1.)Tan[(x^2)]\[DifferentialD]x

NIntegrate[Tan[(x^2)],{x,0,1.}]

xt={0,0.2,0.4,0.6,0.8,1.};
f[x_]:=Tan[(x^2)]
f[xt]

Plot[Abs[f'[x]],{x,0,1}]

Abs[f'[1.]]

0.2(0+0.04+0.16+0.38+0.74)

0.2(0.04+0.16+0.38+0.74+1.56)

Plot[Abs[f''[x]],{x,0,1}]

Abs[f''[1.]]

Plot[Abs[f''[x]],{x,0.9,1}]

(1/24)*49.53047342153081`*(((0.2))^2)

xsr={0.1,0.3,0.5,0.7,0.9};
f[xsr]

0.2(0.1+0.09+0.255+0.533+1.05)

(0.2/2)(0+2*(0.04+0.16+0.38+0.74)+1.56)

Plot[Abs[f''''[x]],{x,0,1}]

Abs[f''''[1.]]

(1/180)9184.473433729345`(0.2)^4

Reduce[(1/180)9184.473433*(1/(n^4))≤ (10^(-5)),n]

Clear[xt,yt]
n=48;
h=1./n;
xt=Table[i*h,{i,0,n}]

f[xt]

n=48;m=n/2;
a=0.;b=1.;
h=(b-a)/n;
xt=Table[a+i*h,{i,0,n}];
IS=(h/3)(f[a]+4(∑_(i=1)^m)f[a+(2i-1)*h]+2(∑_(i=1)^(m-1))f[a+(2i)*h]+f[b])