Integration Problem - Seeking Help to Identify Mistake(s)

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Hello, I have attempted to solve this integration problem ,but I am sure I have gone wrong somewhere.
Can someone please take a look and tell me my mistake(s). Then I will be able to see how to do it correctly.

Thanks kindly for any help.
 

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\int\frac{1}{u}\,du=\ln|u| + C
 
Your problem is that even when you had 1/u du, you still took the integral with respect to x. Take it with respect to u and you should get ln|u|, which you can then back-substitute for x.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

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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