Evaluate the following definite integrals as limit of sums. `int0 4(x+e^(2x))dx`

Explanation

we know that
`overset(b)underset(a)(int)f(x) dx =underset(h to0)("lim") h[f(a)+f(a+h)`
`+f(a+2h)+.....+f{a+(n-1)h}]`
where nh=b-a
`" For " overset(0)underset(4)(int) (x+e^(2x))dx`
here a=0,b=4 and nh =4 and `f(x) =(x|^(2x))`
`:. overset(4)underset(0)(int) f(x) dx=underset(h to0)("lim") h[f(0)+f(0+h)`
`+f(0+2h)+......+f{0(n-1)h}]`
`rArr overset(4)underset(4)(int)(x+e^(2x))dx=underset(h to0)("lim") h[1+(+e^(2h))+(2h+e^(4h))`
`+ {1+e^(2h)+e^(4h)+......+e^(2(n-1)h)}]`
`=underset(h to0)("lim") h[h{1+2+3+......+(n-1)}`
`+{1+e^(2h)+e^(4h)+.....+e^(2nh-2h)}]`
`=underset(h to0)("lim") h[h((n-1)n)/(2)+h{((e^(2h))^(n)-1)/(e^(2h)-1)}]`
`=underset(h to0)("lim") {h^(2)((n-1)n)/(2)+h((e^(2nh)-1))/(e^(2h)-1)}`
`=underset(h to0)("lim") ((nh-h)(nh))/(2)+underset(h to0)("lim") ((e^(2))^(4)-1)/((e^(2h)-1)/(2h)xx2)`
`=underset(h to0)("lim") {((4-h)4)/(2)+(e^(8)-1)/(2)}( . :. underset(x to0)("lim") (e^(x)-1)/(x)=1)`
`=(4xx4)/(2)+(e^(8)-1)/(2)`
`=(8-(1)/(2))+(e^(8))/(2)=(15+e^(8))/(2)`