`int_(0)^(5)(x+1)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=0,b=5" और "nh=5-0=5`
और `" "f(x)= (x+1)`
`therefore int_(0)^(5)(x+1)dx=underset(hrarr0)(lim)h[1+(1+h)+(1+2h)+...+{1+(n-1)h}]`
`=underset(hrarr0)(lim) h[1+1+1+...+" बार "+h{1+2+...+(n-1)}]`
`=underset(hrarr0)(lim)h[n+h(n(n-1))/(2)]`
`=underset(hrarr0)(lim)[nh+h^(2)(n(n-1))/(2)]`
`=underset(hrarr0)(lim)[hn+((nh-h)(nh))/(2)]`
`=underset(hrarr0)(lim)[5+((5-h)(5))/(2)]" "` ( यहाँ , nh = 5)
`=underset(hrarr0)(lim)[5+(5xx5)/(2)=(35)/(2)]`