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

Explanation

`int_{1}^{3}(x^{2}+3 x) d x` `rArr a=1, b=3, f(x)=x^{2}+3 x, h=frac{2}{n}`


`I=int_{1}^{3}(x^{2}+3 x) d x`

`=lim _{h rightarrow 0} h[f(1)+f(1+h)+f(1+2 h)+ldots+f(1+(n-1) h)]`

`=lim _{h rightarrow 0} h[4+((1+h)^{2}+3(1+h)]+[(1+2 h)^{2}+3(1+2 h)]+ldots+[(1+(n-1) h)^{2}+3(1+(n-1) h)]`


`=lim _{h rightarrow 0} h[4+(n-1) cdot 1+h^{2}(1+2^{2}+3^{2}+ldots(n-1)^{2})+2 h(1+2+3+(n-1))]+3(n-1)+3 h(1+2+3 ldots(n-1)) `

`=lim _{h rightarrow 0} h[4 n+h^{2} frac{(n-1)(n-2)(2 n-3)}{6}+2 h(frac{n-1(n)}{2})+3 h(frac{(n-1)(n)}{2}) ]`


`=lim _{n rightarrow infty} 4 n cdot frac{2}{n} +frac{8}{n^{3}} frac{(2 n^{3} ldots)}{6}+frac{5.4}{n^{2}}(frac{n^{2}-n}{2}) ]`


`=8+0+frac{8}{3}+10 )`

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