Integration of even powers of sine and cosine

lydia_y620
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Homework Statement


upload_2017-11-7_14-14-38.png


Homework Equations


cos2x = (1+cos2x)/2
sin2x = (1-cos2x)/2

The Attempt at a Solution


I believe you would use the double angle formula repeatedly but that is very tedious; is there a more concise way to solve the problem?
 

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What about using ##cos^2x + sin^2x =1##?
 
PeroK said:
What about using ##cos^2x + sin^2x =1##?
okay, I've figured it out. Thanks!
 
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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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