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

Explanation

Given: `int_1^3(3x-2)dx`

`I=int_a^b f(x)dx=(b-a)lim_(n->oo)(1/n)[f(a)+f(a+h)+......+f(a+(n-1)h)]`

Here, `a=1,b=3, f(x)=3x-2, h=(b-a)/n=(3-1)/n=2/n`

`I=lim_(n->oo)(1/n)[f(1)+f(1+h)+......+f(1+(n-1)h)]`

`I=lim_(n->oo)(2/n)[3(1)-2+3(1+h)-2+3(1+2h).......+f(1+(n-1)h]`

`I=lim_(n->oo)(2/n)[3-2+3+3n-2+3+3(2)n-2......3+3(n-1)h-2]`

`I=lim_(n->oo)[(2/n)(3(n)-2(n)+3n(1+2+......n-1))]`

Then

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