Solving Log Algebra: Understanding Integrals with (2/2-z) + (2/1+2z)dz

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half way through a question and I'm stumped, I've got all the answers and I'm just revising so the answers no problem just don't get how the integral of {(2/2-z) + (2/1+2z)}dz become -2ln|2-z|+ln|1+2z| +c
 
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Do the two terms separately. For 2/(2-z) use the u substitution u=2-z. What substitution would you use for the second term?
 

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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