Can I use the power rule to integrate a square root in the denominator?

Question

$$\int { \frac { dx }{ \sqrt { 3-4{ x }^{ 2 } } } } $$

Can I use the power rule to integrate this integral if first I transform the $ \sqrt{(3-4x^2)}$ to $(3-4x^2)^{-1/2}$$?$ Which techniques can I use?

Answer 1

Can I use the power rule to integrate this integral if first I transform the $1/ \sqrt{(3-4x^2)}$ to $(3-4x^2)^{-1/2}$?

You can't obtain in general the antiderivative of $$ \frac1{\sqrt{u}} $$ you need a differential element $$ \frac{u'}{\sqrt{u}} $$ here $$ (3-4x^2)'=-4x\ne1. $$ Another route is better.

Answer 2

This is a very natural question. The short answer is no.

But there is a better answer to give you, which is that you are capable of checking your antiderivatives. Namely, if you believe that $$ \int (3 - 4x^2)^{-1/2} dx \stackrel{?}{=} 2(3 - 4x^2)^{1/2},$$ then you can check this by verifying that $$ \frac{d}{dx} 2 (3 - 4x^2)^{1/2} = (3 - 4x^2)^{-1/2}.$$ (Or if you think the antiderivative is a little bit different, you can check that one instead). Perfoming that check here, you will see that this relationship does not hold.