Question

How do I find the anti derivative of `(2x^2+5)/(x^2+1)`?

Answer 1

Rewrite the integrand.

Explanation:

`(2x^2+5)/(x^2+1) = (2x^2+2+3)/(x^2+1)`

`= (2x^2+2)/(x^2+1)+3/(x^2+1)`

`= 2 + 3(1/(x^2+1))`

So the antiderivative is

`2x+3tan^-1x +C`

If you haven't memorized `int 1/(x^2+1) dx = tan^-1x+C`, then you can use trigonometric substitution

`x = tan theta` so `dx = sec^2theta d theta` (and `x = tan^-1 theta`)

After substitution, the integral becomes

`int 1/(tan^2theta + 1) sec^2theta d theta = int 1/sec^2 theta sec^2theta d theta`

`= int 1 d theta = theta +C = tan^-1x +C`

If you don't notice how to split the numerator then do the division.

`(2x^2+5) -: (x^2+1) = 2 + 3/(x^2+1)`

When finding the antiderivative (integral) of a rational function, if the degree of the numerator is greater than the degree of its denominator, division is often a good idea.