Evaluate the following integrals as limit of sum: `int_1^3(2x+3)dx`

Explanation

`I=int_1^3(2x+3)dx`
we know,
`int_a^b f(x)dx =(b-a) lim_(n->oo)1/n[f(a)+f(a+h)+f(a+2h)+....+f(a+(n-1)h)`
where `h=(b-a)/h`
so,`int_a^b f(x)dx =(b-a) lim_(n->oo)1/n sum_(k=0)^n f(a+kh)`
`f(x)=(2x+3)`
`a=1,b=3`
`h=2/n`
`i=2* lim_(n->oo)1/n sum_(k=0)^n f(1+(2k)/n)`
`=2 lim_(n->oo)1/n [sum_(k=0)^n (2 xx (1+(2k)/h)+3)]`
`= 2lim_(n->oo)1/n [sum_(k=0)^n 5((4k)/h]`
`I=2 lim_(n->oo)1/n [(sum_(k=0)^n s)+4/n (sum_(k=0)^n k)]`
`I=2 lim_(n->oo)1/n[5(n+1)+4/n xx (n(n+1))/2]`
`=2 lim_(n->oo)1/n[5n+5+2n+2]`
`=2 lim_(n->oo)1/n(7n+7)`
`=2 lim_(n->oo) 7[(n+1)/n]`
`=14 lim_(n->oo) (1+1/n)`
`I=14 xx 1`
`=14`