MYTUTOR SUBJECT ANSWERS

1001 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

271 SUBJECT SPECIALISTS

£30 /hr

Mathew V.

Degree: Mathematics (Bachelors) - Warwick University

Subjects offered:Maths, Physics+ 1 more

Maths
Physics
Further Mathematics

“Hi there! I’m Mathew, a 3rd year undergraduate student of Mathematics at Warwick University. For A-Levels I studied Maths, Further Maths, Further Additional Maths and Physics. I also worked at a local tuition centre for 2 years prior ...”

£24 /hr

Joe C.

Degree: General Engineering (Masters) - Durham University

Subjects offered:Maths, Science+ 2 more

Maths
Science
Physics
Computing

“Masters student at Durham University, ready to help you excel at Maths and Science ”

Aisling R. A Level Maths tutor, A Level Chemistry tutor
£24 /hr

Aisling R.

Degree: Chemistry (Bachelors) - Bristol University

Subjects offered:Maths, Chemistry

Maths
Chemistry

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

About the author

Samuel C.

Currently unavailable: until 02/06/2017

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

Given that y = 5x^2 - 4/(x^3), x not equal to 0, find dy/dx.

What is a Tree Diagram?

Differentiate the following equation with respect to x; sinx + 3x^2 - 2.

How do I find the turning points of a curve?

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