Evaluate the following definite integrals as limit of sum: `int_(-1)^1(e^(2x)dx)`

Explanation

`int_{a}^{b} f(n) cdot d x=frac{b-a}{n} lim _{n rightarrow infty}[f(a)+f(a+h)+ldots+f(a+(n-1) h]`

`h=frac{b-a}{n}`

`f(x)=e^{2 x}` `a=-1, b=1` , `h=frac{1+1}{n}=2 / n`
`int_{-1}^{1} e^{2 x} cdot d x=frac{2}{n} lim _{n rightarrow infty}[e^{2 a}+e^{2(a+h)}+e^{2(a+2 h)}+ldots+e^{2[a+(n-1) n]}]`

`=frac{2}{n} lim _{n rightarrow infty}[e^{2 a}+e^{2 a+2 h}+e^{(2 a+4 n)}+ldots+e^{2 a+2(n+1)}]`


`=frac{2}{n} lim _{n rightarrow infty}[e^{2 a}+e^{2 a} cdot e^{2 n}+e^{2 a} cdot e^{4 h}+cdots+e^{2 q} cdot e^{2(n-1) h}]`

`=frac{2}{n} lim _{h rightarrow infty}[e^{2 a}(1+e^{2 h}+e^{4 n}+ldots+e^{2(n-1) h}]`

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