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# Integration by parts: ∫ln(x)dx

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.E (LO)
,
FUN‑6.E.1 (EK)
Worked example of finding an indefinite integral using integration by parts, where the integrand isn't a product. Created by Sal Khan.

## Want to join the conversation?

• f(x) = 1/ln(x),

is there a vertical asymptote at x=0 ?

I think that there is...is there?

Also, is the point x=1 a global minima?
I think not because ln(1) = 0 and therefore f(x) is not define there so it can't be minima.

What do you think?

Thanks.
• I don't think there is a vertical asymptote at x=0 because lim(x-->0+) f(x) = 0
• When you differentiate the end result, don't you get ln(x)-1 rather than ln(x)?
• The calculation follows the chain rule : d/dx (x ln x ) = 1 * ln x + x * 1/x = ln x + 1
So, in d/dx (x ln x - x) you have to add d/dx (-x) = -1
Together : = ln x + 1 - 1 = ln x
• Can xlnx be written as ln x^2 ?
If not then why ?
• If you are trying to use properties of logarithms, you would bring the "x" from the front into the logarithm as an exponent, resulting in:

ln (x^x)
• At sal say the word integrand. What is the difference between an integral and an integrand?
• An integral is the whole operator: ∫ f(x) dx
An integrand is just the function you are integrating. So for ∫ 3x^2 dx, the integrand is 3x^2.
• At he integrated g'(x)=1 to get g(x)=x, but shouldn't the integral of g'(x)=1 be g(x)=x+c?
• That is correct if that is where you were going to end your problem, but since there will be further integrals down the road, you can just add a +C at the end of the problem to encompass all the +C you would have had to put in.
• why do you consider 1 as a function in this case and not in other cases?
• You are going to see more and more of this if you continue in math, that is, the creative use of the rules and properties of numbers and processes. In this case, treating the 1 as the result of differentiating some function g(x)=x, made it possible the use of integration by parts to solve the problem. Use whatever works to solve problems. Get creative. But stay within the rules. For me, this is the most fun part of math where you can unleash your creativity! At its best, it is the playground of new ideas, at its worst, it is where you hone your intuition by learning what works and what doesn't - and that isn't bad at all!
• Hi, just doing some revision for this, I always thought that the integral of Ln(x) was always 1/x?
• No, the derivative of ln(x) is 1/x. As Sal points out here, ∫ lnx dx is
xlnx-x+c
• How do I know which part of the function is f(x) and which is g'(x)? I always end up trying both possibilities.
• You need to develop an intuition for which function will simplify with either taking the derivative or the anti-derivative. It's a matter of practice.