योग परिभाषा द्वारा `int_(1)^(3)(x^(2)+1)dx` का मान ज्ञात कीजिए।

Explanation

`int_(1)^(2)(x^(2)+x)dx`
यदि a = 1, b = 3, nh = 2 व `f(x)=(x^(2)+x),` तब,
`int_(1)^(3)(x^(2)+x)dx=underset(hrarr0)(lim)hf(1)+f(1_h)+f(1_2h)+…+f{1+(n-1)h}]`
`=underset(hrarr0)(lim)h[(1^(2)+1)+{(1+h)^(2)+(1+h)}+{(1+2h)^(2)+(1+2h)}+…+{1+(n-1)h}^(2)+{1+(n-1)h}]`
`=underset(hrarr0)(lim)h[2n+h^(2){1^(2)+2^(2)+...+(n-1)^(2)}+3h{1+2+...+(n-1)}]`
`=underset(hrarr0)(lim)[2n+h^(2).((n-1)n(2n-1))/(6)+3h.(n(n-1))/(2)]`
`=underset(hrarr0)(lim)[2nh+((nh-h)nh(2nh-h))/(6)+(3)/(2).nh(nh-h)]`
`=underset(hrarr0)(lim)[4+((2-h)2(4-h))/(6)+(3)/(2).2(2-h)]` जहाँ `nh=2`
`=(38)/(3)`