Integration (core 3)

TheCenturion · 10 Jun 2011 at 13:24

Hello guys, I was wondering if you would be able to help me.

Integrating x/2x+1

Using the substitution u = 2x + 1

I've got to the point:

x/u dx

I don't know what to do from here. presumably I need to differentiate u to just 2 and somehow work du into the integration?

Thanks

Brian8bit · 10 Jun 2011 at 13:26

From the thread title I thought this was going to be a thread ranting about immigrants. That's what I've come to expect of GD.

TheCenturion · 10 Jun 2011 at 13:27

Brian8bit said:
From the thread title I thought this was going to be a thread ranting about immigrants. That's what I've come to expect of GD.

Duly noted, and thread title changed

Mason- · 10 Jun 2011 at 13:59

EDIT: Realised I didn't explain much

you have x/u dx you need to replace all parts of x. So the (x) part and the (dx) part

x can be got from rearranging u = 2x+1 -> x = (1/2)(u-1)

get the dx from doing, du/dx = 2 -> du = 2dx -> (1/2)du = dx

Then sub in to the equation, so
(1/2)(u-1)/u (1/2)du

take both the halfs onto the denominator and you get

(u-1)/4u du

Then separate it into two terms

(u/4u) - (1/4u) du
(1/4) - (1/4u) du

Integrate both terms

(1/4)u - (1/4)ln(u) + c (constant of integration)

sub u = 2x + 1 back in

(1/4)(2x+1) - (1/4)ln(2x+1) + c

take a factor of a quarter out

(1/4)(2x + 1 - ln(2x+1)) + c

and there you go.

You got exams coming up? C3 + C4?
I'm doing further maths A-Level at the moment second year, need any more help with anything just ask.

Squakingcow · 10 Jun 2011 at 14:33

I wouldn't use a substitution to solve that (it makes things way more complex than they need to be). Simply split it into two fractions, which will leave you with it in a form that can easily be integrated.

(1/2)-1/(4x+2)

TheCenturion · 10 Jun 2011 at 14:38

Mason- said:
Snip

Thank you so much mate. You're a saviour

Understand it now, finally

I'm in first year (AS Levels) at the moment, but doing further maths so we're doing C3 early. I got C3, D1 and S2 coming up.

Squakingcow said:
I wouldn't use a substitution to solve that (it makes things way more complex than they need to be). Simply split it into two fractions, which will leave you with it in a form that can easily be integrated.

(1/2)-1/(4x+2)

I probably would've done something like that, but the question specifically said to use substitution.