How Can Multiplying by a Form of 1 Simplify Integrating 1/(1-sec x)?

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Homework Statement



Integrate 1/(1-secx)dx by multiplying the integrand by a form of 1.


Homework Equations



any sort of trig identity?

The Attempt at a Solution



I've tried multiplying by forms of 1 such as secx+tanx, secx-tanx, and others, but when i do, i just end up with a messy looking integrand.
any pointers with this and any other problems like it?
 
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How about multiplying by (1 + sec x)/(1 + sec x)? Is that one you tried?
 

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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