Evaluate: `int_1^3(2x^2+5x)dx` as limit of a sum

Explanation

As We know that,



`int_{1}^{3}(2 x^{2}+5)d u =lim _{h rightarrow 0} h sum_{r=1}^{n} 2(1+r h)^{2}+5 `, where ,` x h=3-1=2`

`=lim _{h rightarrow 0} h sum_{r=1}^{n} 2(1+2 r h+r^{2} h^{2})+5`



`=lim _{h rightarrow 0} h sum_{r=1}^{n} 7+4 r h+2 r^{2} h^{2}`

`=lim _{h rightarrow 0} h[(7+4 h+2 h^{2})+(7+4 h cdot 2+2 h^{2} cdot 2^{2})+.
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