MYTUTOR SUBJECT ANSWERS

566 views

Find the value of 2∫1 (6x+1) / (6x2-7x+2) dx, expressing your answer in the form mln(2) + nln(3), where m and n are integers.

This fraction can’t be integrated easily, but if we split it using partial fractions, these will be easier to integrate. To do this, we need to factoise the denominator, as this will follow the method of partial fractions:

21 (6x+1) / (6x2-7x+2) dx = 21 (6x+1) / ((3x-2)(2x-1)) dx

We can then use partial fractions to split this fraction into two that can be integrated, by using the variables A and B to represent expressions that would multiply together to make 6x+1 when put above the parts of the factoised denominator.

21 (6x+1) / ((3x-2)(2x-1)) dx = 21 A/(3x-2) + B/(2x-1)dx

Following the method of partial fractions, cross-multiply the fractions:

21 A/(3x-2) + B/(2x-1)dx = 21 A(2x-1)/(3x-2) + B(3x-2)/(2x-1) dx

                                        = 21 (2Ax – A + 3Bx – 2B)/((3x-2)(2x-1)) dx

Therefore, we can say 2Ax – A + 3Bx – 2B = 6x+1, and so 2Ax + 3Bx = 6x and -A -2B = 1. We can then solve these simultaneous equations by elimination or substiution and find that B = -8 and A = 15. Therefore:

21 (6x+1) / ((3x-2)(2x-1)) dx = 21 A/(3x-2) + B/(2x-1) dx

                                             = 21 15/(3x-2) + -8/(2x-1) dx

                                             = 21 15/(3x-2) dx - 21 8/(2x-1) dx

Using standard integrals, the integral of a fraction where the numerator is the derivative of the denominator is ln|denominator|. The fractions above are almost like this, if we rewrite them as:

21 15/(3x-2) dx - 21 8/(2x-1) dx = (5)21 3/(3x-2) dx - (4)21 2/(2x-1) dx

                                                   = 5[ln|3x-2|]21 – 4[ln|2x-1|]21

                                                   = 5(ln(4) – ln(1)) – 4(ln(3) – ln(1))

                                                   = 5ln(4) – 4ln(3)

To get this into the form required for the question, we can use the law of logs: log(ab) = b log(a):

5ln(4) – 4ln(3) = 5ln(22) – 4ln(3)

                       = 5(2)ln(2) – 4ln(3)

                       = 10ln(2) - 4ln(3)

Jed M. GCSE Biology tutor, A Level Biology tutor, GCSE Chemistry tuto...

11 months ago

Answered by Jed, an A Level Maths tutor with MyTutor


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

371 SUBJECT SPECIALISTS

£20 /hr

Dan W.

Degree: Economics and Accounting (Bachelors) - Bristol University

Subjects offered:Maths, Economics

Maths
Economics

“I achieved top grades whilst juggling cricket at a high level. I’ve tutored for Young Einstein Tuition & been a Peer Mentor to those facing personal issues”

£20 /hr

Ryan A.

Degree: Mathematics (Bachelors) - Warwick University

Subjects offered:Maths, Further Mathematics + 1 more

Maths
Further Mathematics
.STEP.

“Passionate about passing down a greater understanding of maths and instilling confidence in all students.”

MyTutor guarantee

|  3 completed tutorials

PremiumGiulio P. GCSE Maths tutor, A Level Maths tutor, A Level Economics tu...
£36 /hr

Giulio P.

Degree: Mathematics (Masters) - Bristol University

Subjects offered:Maths, Physics+ 3 more

Maths
Physics
Italian
Further Mathematics
Economics

“Exam Time! I can provide a rapid revision class in any maths module which will test your son or daughter in all the fundamentals”

About the author

Jed M.

Currently unavailable: for regular students

Degree: Natural Sciences (Bachelors) - Exeter University

Subjects offered:Maths, Chemistry+ 1 more

Maths
Chemistry
Biology

“Hi! I'm a second year Natural Science student at the University of Exeter. I love all things science, and enjoy sharing what I know about it with others. For my GCSEs and A Levels, I found worked examples and practice questions are ke...”

You may also like...

Other A Level Maths questions

What is a parametric equation?

A line has an equation y = e^(2x) - 10e^(x) +12x, find dy/dx

How do you differentiate e^(x^2) ?

Why maths is so hard sometimes?

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