Differentiation, I'm I doing it right?

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I took dt of both sides. The part that confuses me is when you take the derivative of 1 over t . The one becomes a 0 and the t becomes a dt. Just want to make sure I did that right.
 

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If you are taking the derivative of 1/t think of it as ##t^{-1}## and apply the rule for powers.
 
So it would be , -t^-2?
 
Edit: crossposted. You are on the right track now.

You should not end up with ## dt ## on its own. Go back to your notes and look at how to differentiate two terms added together, and how to differentiate ## x^n ## (bearing in mind that ## x = x^1 ## and ## \frac 1x = x^{-1} ##).
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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