integral with variable as limit, and proving conditions

Question

Ok, so I am working an a calculus assignment. We are working with integrals, and I am given the task:

Let the function L be defined as:

$$L(x) = \int_{1}^{x} \frac{1}{t}dt$$

I am not allowed to use $L(x) = ln(x)$. then the task is:

Prove that for all $x,y >0$ the following is true:

i) $L(xy) = L(x) + L(y)$

ii) $L(\frac{1}{x}) = -L(x)$

I got a tips from another student that I should derivate the function L, and that brings me to my question. How do I work with integrals that have a variable as a limit, and how do i derivate them? I am sure some of our courses has covered this, but I cannot even remotely remember how.

Answer 1

HINTS (try yourself first before looking)

For part a, substitution of $u=t\cdot x$ will work.

For part b, substitute $u=1/t$.

For the comment of your fellow student, look into the Fundamental Theorem of Calculus.