`int_(1)^(4)(x^(2)-x)dx`

Explanation

हम जानते हैं
`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`
दिया हैं, `int_(1)^(4)(x^(2)-x)dx,` यहाँ `a=1, b=4`
तथा nh = 3 और `f(x)=x^(2)-x=x(x-1)`
`int_(1)^(4)(x^(2)-x)dx=underset(hrarr0)(lim)h[f(1)+f(1+h)+f(1+2h)+...+f{1+(n-1)h}]`
`=underset(hrarr0)(lim)h[(1-1)+(1+h)h+(1+2h)(2h)+...+{1+(n-1)h}{(n-1)h}]`
`=underset(hrarr0)(lim)h^(2)[(1+h)+2(1+2h)+3(1+3h)+...+(n-1){1+(n-1)h}]`
`=underset(hrarr0)(lim)h^(2)[{1+2+3+...+(n-1)}+{h+2^(2)h+3^(2)h+...+(n-1)^(2)h}]`
`=underset(hrarr0)(lim)h^(2)[((n-1)n)/(2)+h{1^(2)+2^(2)+...+(n-1)^(2)}]`
`=underset(hrarr0)(lim){((n-1)nh^(2))/(2)+(h^(3)(n-1)n(2n-1))/(6)}`
`=underset(hrarr0)(lim){((nh-h)(nh))/(2)+((nh-h)(nh)(2nh-h))/(6)}`
`=underset(hrarr0)(lim){((3-h)3)/(2)+((3-h)3(2xx3-h))/(6)}`
`=(9)/(2)+(3xx3xx6)/(6)=(9)/(2)+9=(27)/(2)`