Evaluate the following integrals as a limit of sums:

Formula used:

where,

Here, a = 1 and b = 3

Therefore,

Let,

Here, f(x) = 3x – 2 and a = 1

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

f(1) = 3(1) – 2 = 3 – 2 = 1

Similarly, f(1 + h)

= 3(1 + h) – 2

= 3 + 3h – 2

= 3h + 1

In this series, 1 is getting added n times

Now take 3h common in remaining series

Put,

Since,