Integrate (x+4)/(x^2+2x+2)

At first glance, this may look like an inverse trig integral but as the top contains an x term, we must use a different method.

First, rewrite the top in two parts, one that is a multiple of the derivative of the bottom and one that is just a number. In this case the derivative of the bottom is 2x+2 so rewrite as (2x+2)/2+3.

Now split the fraction in two: (2x+2)/2(x2+2x+2)+3/(x2+2x+2). The first part is now a log integral integrating to 1/2 ln(x2+2x+2).

To integrate the second we must complete the square on the bottom. x2+2x+2=(x+1)2+1. Now using the substitution u=x+1, dx=du, we can see that the second part is an inverse tan integral integrating to arctan(x+1).

The overall integral is therefore 1/2 ln(x2+2x+2) + arctan(x+1).

Zac T. A Level Chemistry tutor, A Level Further Mathematics  tutor, A...

7 months ago

Answered by Zac, an A Level Further Mathematics tutor with MyTutor

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


£20 /hr

Solomon L.

Degree: Mathematics (Bachelors) - Leeds University

Subjects offered: Further Mathematics , Maths

Further Mathematics

“Currently studying at Leeds university, ready to help you improve your grades in Maths and Chemistry GCSE! ”

£22 /hr

Costas Y.

Degree: Electrical & Electronic Engineering (Doctorate) - Imperial College London University

Subjects offered: Further Mathematics , Maths+ 1 more

Further Mathematics

“About Me: I am a 3rd year PhD student at Imperial College London studying in the Electrical and Electronic Engineering department. Ambitious and enthusiastic engineer and have a passion for mathematics and electronics. Aside from my...”

MyTutor guarantee

£30 /hr

Thomas D.

Degree: Economics (Bachelors) - Bristol University

Subjects offered: Further Mathematics , Maths+ 2 more

Further Mathematics

“Why choose me? I am a highly academic student having achieved 4 A* at A level but also have excellent communication and mentoring skills havingtutored fellow students at school and beenHead Boy. I am honest, reliable and precise, if ...”

About the author

Zac T.

Currently unavailable: until 03/10/2016

Degree: Natural Sciences (Bachelors) - Cambridge University

Subjects offered: Further Mathematics , Science+ 4 more

Further Mathematics

“About me I am a student at the University of Cambridge where I study Natural Sciences, a course which allows me to maintain my interest in multiple areas of science and maths. I take modules in Chemistry, Physics, Earth Science and Ma...”

MyTutor guarantee

You may also like...

Posts by Zac

Find the shortest distance between the line L: x=1+t, y=1+2t, z=1-t and the point A: (2,3,4)

How can the average speedx of a gas molecule be derived?

Integrate (x+4)/(x^2+2x+2)

Why are transition metal complexes coloured?

Other A Level Further Mathematics questions

Find the complex number z such that 5iz+3z* +16 = 8i. Give your answer in the form a + bi, where a and b are real numbers.

Prove by mathematical induction that, for all non-negative integers n, 11^(2n) + 25^n + 22 is divisible by 24

Find the inverse of the general 2x2 matrix A= ([a, b],[c, d]) when does this inverse exist?

How does proof by induction work?

View A Level Further Mathematics tutors


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