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

Explanation

हम जानते हैं कि
`int_(a)^(b)f(x)dx=underset(hrarr0)(lim)h[f(a)+f(a+h)+f(a+2h)+…+f{a+(n-1)h}]`
जहाँ,`" "nh=b-a`
यहाँ,`" "a=a,b=b" और "f(x)=x`
`therefore int_(a)^(b)x dx = underset(hrarr0)(lim)h[a+(a+h)+(a+2h)+…+a+(n-1)h]`
`=underset(hrarr0)(lim)h[(a+a+...+n" बार ")+h{1+2+...(n-1)}]`
`=underset(hrarr0)(lim)h[na+h{1+2+3+...+(n-1)}]`
`=underset(hrarr0)(lim)[hna+h^(2)(n(n-1))/(2)]`
`=underset(hrarr0)(lim)[hna+(nh(nh-h))/(2)]`
`=underset(hrarr0)(lim)[(b-a)a+((b-a)(b-a-h))/(2)](because nh=b-a)`
`=[(b-a)a+((b-a)^(2))/(2)]`
`=(b-a)(a+(b-a)/(2))=(b-a)((2a+b-a)/(2))`
`=(b-a)((a+b)/(2))=(b^(2)-a^(2))/(2)`