Guide for what the terms mean:
ln(x) - "ln" is a logarithm of base e.
logarithm (from the definition of ln(x) ) - logarithm is a function, basically we have multiplication and division and we also have logarithms and powers. Lets say Log(x) is logarithm to the base 10 of a value x. 10^(log(x)) is equal to x. log (1000) is 3 because 10^3 is 1000.
e (from the definition of ln(x) ) - e is an important number in maths, like pi. It has infinite decimal places, like pi. It is roughly 2.718
dx - Differentiation is another function. When you differentiate x^2 you take the number in the power (2 here) and multiply by that, then you take 1 away from the power, then add "dx" to the end. If you differentiate u^2 it's 2u du. So differentiating x^2 = 2x dx. In basic terms, dx is just something you add to the end of the equation after differentiating.
Integration - Integration is the opposite of differentiation. Integrating a number, we add 1 to the power (the power of the number beside the d) and then divide by the number in the power. E.g. integrating x^2, we get (x^3)/3 and then we remove the dx or d(whatever letter we're integrating with respect to)
Now:
u = ln(x)
On the left, we're differentiating u, so we multiply by the power (which is just 1, here) then take away 1 from the power, leaving us with u^0. Any number to the power of 0 is just 1. We're left with 1 * du, or just du.
On the right, we have ln(x). Some values, when differentiated have different rules than just taking one from the power. For a logarithm, the differentiation is 1/x giving us 1/x dx
So, differentiating u = ln(x) we get du = 1/x dx
The bottom equation, we're integrating.
dv is really 1*dv, so integrating 1 (with respect to v because of the dv) we just get v and remove the dv.
x^5 dx, we add 1 to the x and divide by the number in the power, to get (x^6)/6
Yes, but In this case, it looks like it's just some side work for an equation where integration by parts is needed. If that's the case, adding the constant in there is unnecessary because there will be a further integration. Since the constant is arbitrary, you can keep it until you've integrated fully.
For integration, you add one to the power and divide by whatever the power is. So x^5, you add 1 and it becomes x^6 and then you divide by 6 to give you (x^6)/6