Evaluate the following integral as a limit of sums: ∫ (3x^2 + 5x)dx, x ∈ [0, 1]

Question

Evaluate the following integral as a limit of sums:

\(\int\limits_{0}^{1} \)(3x² + 5x)dx

Answer 1

To find: \(\int\limits_{0}^{1} \)(3x² + 5x)dx

Formula used:

where,

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 3h 2 and 5h common in remaining series

Put,

h = \(\cfrac1n\)

Since,

Hence, the value of 

 \(\int\limits_{0}^{1} \)(3x² + 5x)dx = \(\cfrac72\)