Evaluate: `int_a^b e^x dx` using limit of sum

Explanation

`"Let "f(x) =2x+3`
`"Here"" "a=1,b=3`
`:. " "nh=b-a=3-1=2`
`"Now" int_(1)^(3)f(x) dx=underset(hto0)(lim)h[f(1+h)f(1+2h)`
` +f(1+3h) +.....+ f(1+nh)]`
`underset(hto0)("=lim") h[{(5+2h) +(5+4h) +...+(5+2nh)]`
`underset(hto0)("=lim") h[5n+2h(1+2+3+.....+n)]`
`underset(hto0)("=lim") h[5n +2h . (n(n+1))/(2)`
`underset(hto0)("=lim") [5nh +nh (nh +h)]`
`underset(hto0)("=lim") [5(2)+2(2+h)]`
`=10 +4 =14`