`int_(a)^(b) x dx`

Explanation

we know that
`overset(b)underset(a)(int) f(x) dx= underset( h to0)(" lim") h [f(a)+f(a+h)`
`+f(a+2h)+....+f{a+(n-1)h}]`
` " where " nh=b-a`
` " Here " a=a,b= b " and " f(x)=x`
` :.underset(a)overset(b)(int) x dx =underset( h to 0)("lim") h[a+(a+h)+(a+2h)`
`+.....+a+(n-1)h]`
`underset(h to0)("lim") h[(a+a+.....+ n " times ")`
`+h{1+2+.....(n-1)}]`
`=underset(h to 0)("lim") [hna +h^(2).(n(-1))/(2)]`
`=underset(h to 0)("lim") [hna +.(hn(nh-h))/(2)]`
`=underset(h to 0)("lim")[(b-a)a+.((b-a)(b-a-h))/(2)]"(":. nh =b-a")"`
`=[(b-a)a+.((b-a)^(2))/(2)]`
`=(b-a)(a+(b-1)/(2))=(b-a)((2a+b-a)/(2))`
`=(b-a)((a+b)/(2))=(b^(2)-a^(2))/(2)`