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Having a little problem integrating fractions

How do I integrate

(x-1)/(x+1)

2t/(1-t^2)

.. this is probably a very basic question so it doesn't have these in the example questions of the chapter either.. any help is appreciated.

Reply 1

marcusmerehay

The second one is in the form:

\frac{f'(x)}{f(x)}

, which integrates to

log|f(x)|

(a highly useful identity to remember - just don't forget to divide by the appropriate constants as you would when integrating normally).

As for the first one, rewrite it as

\frac{x}{x+1}-\frac{1}{x+1}

. The second fraction follows the same rule as for your second problem, whereas the first one can be integrated using the substitution

x=u-1

(you could alternatively use this substitution to do the whole question in one go).

(edited 12 years ago)

Reply 2

Ziltoid

Original post by rei dos reis

How do I integrate

(x-1)/(x+1)

2t/(1-t^2)

.. this is probably a very basic question so it doesn't have these in the example questions of the chapter either.. any help is appreciated.

For the first one I'd just use integration by parts... let the denominator be one part, and the numerator be another.

For the sencond one a simple substitution of

u = 1-t^2

should do it.

Reply 3

Rational Paradox

For the first question, note that the numerator can be written as

(x + 1) - 2

For the second, its a integral where the function is in the form kf'(x)/f(x)

Reply 4

marcusmerehay

OP - Just to clear up any confusion you may have over this, all methods listed by myself and Ziltoid are viable, as are a great deal of others, it may just be that some answers come out more quickly than others.

I would imagine that if those two problems are next to each other in the same textbook, they are likely asking you to do integration by substitution.