It will be easy if you graph it out.
Note that for x < 1x
< 1, log(x) < 0.
Therefore, for 0 ≤ x ≤ 1, the
area is the area under the curve e^x. For 1 ≤ x ≤, e^x > log(x) ≥.
Putting it together:
∫(x = 0 to 1) e^x dx + ∫(x
= 1 to 4) [e^x - ln(x)] dx
= [e^x] (x = 0 to 1) + [e^x - x(ln(x) - 1)](x = 1 to 4)
= (e^1 - e^0) + ((e^4 - 4(ln(4) - 1)) - (e^1 - 1(ln(1) - 1)))
= (e - 1) + (e^4 - 8 ln(2) + 4 - e + 0 - 1)
= e^4 + 2 - 8 ln(2)
