MYTUTOR SUBJECT ANSWERS

1172 views

How do I find the integral ∫(ln(x))^2dx ?

This problem is all about using integration by parts, so let's start by quoting the formula for integration by parts:

    ∫u*(dv/dx)dx = uv - ∫v*(du/dx)dx

To get the integral we want on the left hand side we can use the subtitutions u = dv/dx = ln(x). This means that we will have to find ∫ln(x)dx, this is also done using integration by parts:

To find ∫ln(x)dx we can use the substitutions u = ln(x) and dv/dx = 1. Using the formula above will then give us:

   ∫ln(x)*1dx = ln(x)*x - ∫x*(1/x)dx

 = xln(x) - ∫dx = xln(x) - x = x(ln(x)-1)

Using this we can now use our original substitutions in the formula to get:

   ∫ln(x)*ln(x)dx = ln(x)*x(ln(x)-1) - ∫x(ln(x)-1)*(1/x)dx

 = xln(x)*(ln(x)-1) - ∫(ln(x)-1)dx 

 = xln(x)*(ln(x)-1) - x(ln(x)-1) + x + c 

Now we just have to tidy this up to get our final answer:

  ∫(ln(x))^2dx = x[(ln(x)+1)^2 + 1] + c

Samuel C. A Level Physics tutor, A Level Maths tutor, A Level Further...

2 years ago

Answered by Samuel, an A Level Maths tutor with MyTutor


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

333 SUBJECT SPECIALISTS

£26 /hr

Ben J.

Degree: Physics and Computer Science (Natural Sciences) (Masters) - Durham University

Subjects offered:Maths, Science+ 3 more

Maths
Science
Physics
Chemistry
-Personal Statements-

“About Me: I'm a Physics and Computer Science student at Durham University, which sounds boring but it's actually really fun.I've got a great knack for making boring things more exciting, which is a good job given my degree title!  I'...”

MyTutor guarantee

£20 /hr

Kate P.

Degree: Mathematics (Masters) - Bath University

Subjects offered:Maths, Geography+ 1 more

Maths
Geography
-Personal Statements-

“Hi! I'm Kate. I'm a mathematics student who strives to improve students confidence, enjoyment and ability in mathematics and geography.”

£20 /hr

Mustafa A.

Degree: Economics (Bachelors) - Birmingham University

Subjects offered:Maths, Economics

Maths
Economics

“Highly proficient Maths and Economics tutor who's goal is to ensure that the student comes from a session with the ability to tackle the problem with ease.”

MyTutor guarantee

About the author

£26 /hr

Samuel C.

Degree: Physics (Bachelors) - Durham University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
Chemistry

“Hi, I'm Sam Crawford. I'm studying Maths and Physics at Durham University, with an offer from Cambridge for next year, and I absolutely love both subjects.”

You may also like...

Posts by Samuel

How do I do a proof by induction?

How do I find the integral ∫(ln(x))^2dx ?

Other A Level Maths questions

What is the natural logarithm?

Find the tangent to the curve y=x^3+3 at the point x=1.

Differentiate with respect to X: x^2 + 2y^2+ 2xy = 2

How do I differentiate sin^2(x)?

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