KIẾM THẾ ORIGIN – SOI HÀNG THIẾU LÂM MẠNH NHẤT TOÀN CÕI SV | LnP
KIẾM THẾ ORIGIN – SOI HÀNG THIẾU LÂM MẠNH NHẤT TOÀN CÕI SV | LnP

## AP®︎/College Calculus BC

• Integration by parts intro
• Integration by parts: ∫x⋅cos(x)dx
• Integration by parts: ∫ln(x)dx
• Integration by parts: ∫x²⋅𝑒ˣdx
• Integration by parts: ∫𝑒ˣ⋅cos(x)dx
• Integration by parts
• Integration by parts: definite integrals
• Integration by parts: definite integrals
• Integration by parts challenge
• Integration by parts review

Integration by parts: ∫ln(x)dx

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?
• I don’t think there is a vertical asymptote at x=0 because lim(x–>0+) f(x) = 0(4 votes)
• When you differentiate the end result, don’t you get ln(x)-1 rather than ln(x)?(12 votes)
• 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(3 votes)
• Can xlnx be written as ln x^2 ?
If not then why ?(0 votes)
• 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:
• Atsal say the word integrand. What is the difference between an integral and an integrand? 2:58(8 votes)
• 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.(12 votes)
• Athe integrated g'(x)=1 to get g(x)=x, but shouldn’t the integral of g'(x)=1 be g(x)=x+c? 2:20(7 votes)
• 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.(5 votes)
• why do you consider 1 as a function in this case and not in other cases?(6 votes)
• 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!(7 votes)
• Hi, just doing some revision for this, I always thought that the integral of Ln(x) was always 1/x?(2 votes)
• No, the derivative of ln(x) is 1/x. As Sal points out here, ∫ lnx dx is
• How do I know which part of the function is f(x) and which is g'(x)? I always end up trying both possibilities.(3 votes)
• 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.(5 votes)
• I am 65 years old, and re-learning calculus on my own, using an old textbook.
When I first come upon integrating ln x, the book has not yet mentioned integration by parts. How would you integrate ln x without using integration by parts?(5 votes)
• Awesome! I’m in my mid 50’s.
Your book probably just provided the cookbook result. I’m not aware of any other method to compute the integral other than IBP.(2 votes)
• If the derivitive of a function is written as the prime of that function (i.e. f'(x)) is there such notation for the antiderivitive of a function?(2 votes)
• The antiderivative is the same as an integral. You can explore this here : https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/indefinite_integrals/v/antiderivatives-and-indefinite-integrals