Question

What is `int1/(x^2+5)` ?

When I use an online integral calculator, it's suggesting I use substitution and let `u=x/sqrt(5)`

I don't understand how to get the `sqrt(5)`. Is there an alternative way of solving this?
If not - what does the `sqrt(5)` mean and how do I work this out for similar questions?

Answer 1

Apologies - I've posted this in the wrong place and can't see how to delete it. :(

Answer 2

(1/sqrt 5) arctan (x / sqrt 5) + C

Explanation:

This is mostly algebraic manipulation and the knowledge that the integral of 1 / ( x^2 + a^2) = 1/a arctan x/a
In this case a = sqrt 5 if a^2 = 5

The substitution comes about to convert x^2 + 5 to u^2 + 1 and using the knowledge that integral of 1 / x^2 + 1 = arctan x

Don't forget that if u = x/(sqrt 5) then dx = (sqrt 5) du