Integrate x/(2-x).?

Question

I have to integrate x/(2-x) dx.
I've tried everything I can think of and I can't figure it out. I think I might be missing something really simple. Or maybe not =\.
Anyway, help is appreciated.
Thanks!

Answer 1

∫x/(2-x) dx

First, rewrite the numerator in terms of x-2:

∫(x-2)/(2-x) + 2/(2-x) dx

Simplify the first fraction:

∫-1 + 2/(2-x) dx

Now integrate (remembering to use the inverse chain rule on the second term):

-x - 2 ln |2-x| + C

Answer 2

Integral Of X 2 X

Answer 3

I took calculus about 10 years ago, I kind of remember...so I think I can put you in the right direction:

=x/(2-x)
=x/(-x+2)
=x/-(x-2)
=-x/(x-2)

then I think you can split the integral (I'll use the S for Integral)

S -x dx S 1/(x-2) dx

which leaves you with

-1/2 x2 ln(x-2) + C

I think...

double check the part where I split the integral...

Answer 4

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

Therefore, integral
= Integral of [ -1 + 2 / ( x - 2) ] dx
= - x + 2 log l x - 2 l + C

Answer 5

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