Evaluate the following definite integrals as limit of sums. `int_0^5(x+1)""dx`

Explanation

we know that
` overset(a)underset(b)(int) f(x) dx=underset(h to 0)("lim") h[f(a+h) +f(a+2h)`
`+....+f{a+(n-1)h}]`
where nh=b-a
Here a,=0, b=5 and nh=5-0=5
and f(x) = (x+1)
`:. overset(0)underset(5)(int) (x+1)dx=underset( h to0)("lim ") h[1+(1+h)`
`+(1+2h)+.....+{1+(n-1)h}]`
`=underset(h to 0)("lim") h[(1+1+1+....+n " times ")`
` +h{1+2+....+(n-1)}]`
`=underset(h to 0)("lim") h[n+h .(n(n-1))/(2)]`
`=underset(h to 0)("lim") [nh +h^(2).((nh-h)(nh))/(2)]`
`=underset(h to 0)("lim") [5+.((5-h)(5))/(2)]`
`= 5+(5xx5)/(2)=(35)/(2)`