Evaluate the following integrals as a limit of sums:

Formula used:

where,

Here, a = 0 and b = 4

Therefore,

Let,

Here, f(x) = x + e^2x and a = 0

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

f(0) = 0 + e²⁽⁰⁾ = 0 + e⁰ = 0 + 1 = 1

f(h)

= h + (e)^2h

= h + e^2h

Similarly, f(2h)

= 2h + (e)^2(2h)

= 2h + e^4h

Take h common in some of the terms of series

Sum of n terms of a G.P. is given by,

and

Therefore,