Evaluate the following integrals as a limit of sums:

Formula used:

where,

Here, a = 0 and b = 1

Therefore,

Let,

Here, f(x) = 3x² + 5x and a = 0

Now, by putting x = 0 in f(x) we get,

f(0) = 3(0)² + 5(0) = 0 + 0 = 0

f(h)

= 3(h)² + 5(h)

= 3h² + 5h

Similarly, f(2h)

= 3(2h)² + 5(2h)

= 3h²(2)² + 5h(2)

Now take 3h² and 5h common in remaining series

Put,

Since,