How do we solve this? I applied the fundamental theorem of calculus and know that f(x) = 1/x and n=b/a, but this is not the correct answer. Can someone please help? Thanks in advance

enter image source here

1 Answer
Apr 30, 2018

ln n = int_1^n 1/x dx," "n>0

Explanation:

You're correct that f(x) = 1/x, because the primitive (anti-derivative) of this function is ln x. Working with the integral on the right side, we get

int_a^b 1/x dx = [ln x + C]_(x=a)^b
color(white)(int_a^b 1/x dx) = [ln b + C] - [ln a + C]
color(white)(int_a^b 1/x dx) = ln b + cancelC - ln a - cancelC
color(white)(int_a^b 1/x dx) = ln b - ln a

We now wish this expression to equal ln n through our choice of a and b.

If we choose a = 1 and b = n, we get

int_1^n 1/x dx = ln n - ln 1
color(white)(int_1^n 1/x dx) = ln n - 0
color(white)(int_1^n 1/x dx) = ln n

This is valid for all n>0.