Question

Find the most general antiderivative of the following function (note I have NOT learned about integrals yet). I'm confused about the chain rule and the antiderivative. The function: 2x+5(1-x^2)^(-1/2)

Answer 1

First of all, for the sake of simplicity, lets pretend that antiderivatives are the same as integrals (they pretty much are).
Also, when I say\[\int\limits_{}{} \frac {2x+5}{\sqrt{1-x^2}}\]
I mean "take the antiderivate of 2x+5(1-x^2)^(-1/2).

So the first thing we notice is that we could rewrite the problem to \[\int\limits_ {}{} \frac {1}{\sqrt{1-x^2}} * 2x+5\]
If we look on the left hand side, we notice that we have the term \[\int\limits_ {}{} \frac {1}{1-x^2}\]
Which is the derivative of inverse sine

\[\sin^{-1}\]

Answer 2

EDIT. I totally mean to have a square root sign under the 1-x^2 up there. My bad.
So from that, we can conclude that inverse sine will have a part in this integral (antiderivative).

So now that we know that inverse sine is part of the question, lets work through the integral again.
\[\int\limits\limits\limits {}{} \frac{1}{\sqrt{1-x^2}} (2x+5)\]
The 1-x^2 is the "driving function," or if you learn u-substitution later, the "U," meaning that if the "driving function/U" is responsible for the (2x+5) term per the chain rule.
If this doesn't make any sense, just wait until you learn U-substitution (or read about it yourself!)