Evaluate the following integrals as a limit of sums:

Formula used:

where,

Here, a = 1 and b = 3

Therefore,

Let,

Here, f(x) = x² + x and a = 1

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

f(1) = 1² + 1 = 1 + 1 = 2

f(1 + h)

= (1 + h)² + (1 + h)

= h² + 1² + 2(h)(1) + 1 + h

= h² + 2h + h + 1 + 1

= h² + 3h + 2

Similarly, f(1 + 2h)

= (1 + 2h)² + (1 + 2h)

= (2h)² + 1² + 2(2h)(1) + 1 + 2h

= (2h)² + 4h + 2h + 1 + 1

= (2h)² + 6h + 2

{∵ (x + y)² = x² + y² + 2xy}

In this series, 2 is getting added n times

Now take h² and 3h common in remaining series

Put,

Since,