MYTUTOR SUBJECT ANSWERS

768 views

How do I integrate ln(x)?

There is a subtle, but very neat trick to this when applying the rules of integration by parts.

If we take ∫ln(x)dx = ∫1*ln(x)dx, and then let our term to be differentiated, u = ln(x), and our term to be integrated, dv/dx = 1, then it follows that:

 

du/dx = x⁻¹, v = x

 

and from the integration by parts formula:

∫u * (dv/dx) dx = uv - ∫v * (du/dx) dx

 

∴ ∫ln(x)dx = xln(x) - ∫(x⁻¹ * x)dx (+ constant)

∫ln(x)dx = xln(x) - ∫dx (+ constant)

Hence, our results turns out to be:

∫ln(x)dx = xln(x) - x + c

 

NB. While our trick here gives us a very straightforward solution to an integration which could have been very laborious via other methods, integration by parts tends to be a last resort, as more, seemingly contrived steps are required. One should generally try integration by substitution, for non-standard integrations, first when unsure of which method to use, as the steps to a result are often far simpler and quicker.

Jacan C. A Level Maths tutor, A Level Physics tutor, GCSE Physics tut...

2 years ago

Answered by Jacan, an A Level Maths tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

369 SUBJECT SPECIALISTS

£30 /hr

Rachel H.

Degree: Physics and Astronomy with a Year Abroad (Bachelors) - Durham University

Subjects offered:Maths, Science+ 3 more

Maths
Science
Physics
German
Chemistry

“Hey, I'm Rachel and I'm Physics graduate from Durham University. My best subjects at school were always science and maths but I also loved languages and art”

£30 /hr

Tadas T.

Degree: MMathPhil Mathematics and Philosophy (Bachelors) - Oxford, St Anne's College University

Subjects offered:Maths, Further Mathematics + 3 more

Maths
Further Mathematics
.MAT.
-Personal Statements-
-Oxbridge Preparation-

“University of Oxford Maths and Philosophy student happy to help students learn and stay motivated!”

£26 /hr

Prina S.

Degree: Environmental Geoscience (Masters) - Bristol University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Geography
Biology

“Hello there! My name is Prina and I love studying and tutoring maths and science. Check out my profile for how I teach and previous students' feedback!”

About the author

PremiumJacan C. A Level Maths tutor, A Level Physics tutor, GCSE Physics tut...
£36 /hr

Jacan C.

Degree: Theoretical Physics (Masters) - York University

Subjects offered:Maths, Science+ 3 more

Maths
Science
Physics
Further Mathematics
Chemistry

“Helping the striving and the struggling with caring, friendly and structured tuition. With 200+ hours of experience, offering Physics, Maths and Chemistry.”

You may also like...

Other A Level Maths questions

Use the Factor Theorem to determine whether x – 1 is a factor of f (x) = 2x^4 + 3x^2 – 5x + 7.

Solve the following simultaneous equations: 3x + 5y = -4 and -2x + 3y = 9

x = 2t + 5, y = 3 + 4/t. a) Find dy/dx at (9.5) and b) find y in terms of x.

I'm supposed to calculate the differential of f(x)= sin(x)*ln(x)*(x-4)^2 using the product rule. I know what the product rule is but I can't split this into two bits that are easy to differentiate. How do I do it?

View A Level Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok