Evaluate the following integrals as limit of sum: `int_0^1(3x^2+5x)dx`

Explanation

Given that,
`I=int_0^1(3x^2+5x)dx`
`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 `a=0,b=1`
`h=(b-a)/h`
`=>h= 1/n`
`=>hn=1`
`int_0^1(3x^2+5x)dx =1 lim_(n->oo) 1/n[f(0)+f(h)+f(2h)+....+f(n-1)h]`
`=lim_(n->oo) 1/n[3(0)^2+5(0)+3h^2+5h+3(2h)^2+5(2h).....3[(n-1)h]^2+5[(n-1)h]]`
`=lim_(n->oo)1/n[3h^2[1^2+2^2.....(n-1)^2]+5h[1+2+3+....(n-1)]]`
`=lim_(n->oo)=1/n[3n^2((n-1(n)(2n-1))/6)+5h((n-1)n)/2]`
`=lim_(n->oo)=1/n[1/n^2[(n(n-1)(2n-1))/2]+5/n(n(n-1))/2]`
`=lim_(n->oo)=1/n^3[(n(n-1)(2n-1))/2]+5/n^2(n(n-1))/2`
`=lim_(n->oo)=((n-1)/n*(n/n)*(2n-1)/n)/2 +5/2 (n/n)*(n-1)/n]`
`=lim_(n->oo)=[((1-1/n)(2-1/n))/2+5/2(1-1/n)]`
`=2/2+5/2`
`=1+5/2`
`=7/2`