(∫_0^0.5)Sin[x]\[DifferentialD]x

(∫_0^0.5)Sin[(E^(x^2))]\[DifferentialD]x

Integrate[Sin[(E^(x^2))],{x,0,0.5}]

NIntegrate[Sin[(E^(x^2))],{x,0,0.5}]

Clear[xt,f]
xt={0.,0.1,0.2,0.3,0.4,0.5};
f[x_]:=Sin[(E^(x^2))]
f[xt]

Plot[Abs[f'[x]],{x,0,0.5}]

(0.5/2)*0.4*0.1

0.1(0.8415+0.8469+0.8628+0.8886+0.9221)

0.1(0.8469+0.8628+0.8886+0.9221+0.9592)

Clear[xsr]
xsr={0.05,0.15,0.25,0.35,0.45};
f[xsr]

{0.8428207928338252`,0.8535466773627736`,0.8745
438795899041`,0.9045482990344064`,0.9406227000825375`}

f[xsr]

SetPrecision[f[xsr],30]

Plot[Abs[f''[x]],{x,0,0.5}]

Abs[f''[0.]]

(0.5/24)1.0806(((0.1))^2)

0.1(.842821+0.8535466773627736`+0.8745438795899041`+0.9045482990344064`+0.9406227000825375`)

(0.1/2)*(0.8414709848078965`+2*(0.8468585254539526`+0.8628143740658407`+0.8885497294056712`+0.9221146924738218`)+0.9591622435375395`)

Plot[Abs[f''''[x]],{x,0,0.5}]

Abs[f''''[0.5]]

Clear[n]
((((0.5-0))^5)/(180(n^4)))86.06789871229572`

Reduce[((0.014942343526440229`)/(n^4))<(10^(-5)),n]

Clear[xt,a,b,h,n]
a=0.; b=0.5; 
n=8; h=((b-a)/n);
xt=Table[a+i*h,{i,0,n}]

Clear[a,b,h,n]
a=0.; b=0.5; 
n=8; h=((b-a)/n);
m=n/2;
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])