Evaluate the following integrals as a limit of sums:

Formula used:

where,

Here, a = 1 and b = 3

Therefore,

Let,

Here, f(x) = 2x + 3 and a = 1

Now, By putting x = 1 in f(x) we get,

f(1) = 2(1) + 3 = 2 + 3 = 5

Similarly, f(1 + h)

= 2(1 + h) + 3

= 2 + 2h + 3

= 2h + 5

In this series, 5 is getting added n times

Now take 2h common in remaining series

Put,

Since,