Question

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

Answer 1

`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`.