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I'm reading this explanation of integrals with quadratics and the author pulled this out of nowhere. Is it obvious to everyone but me that this statement is true?

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    Yes. $$(2\tan^2x + 2)^3 = (2(\tan^2x + 1))^3=2^3(\tan^2x + 1)^3=8(\tan^2x + 1)^3.$$2012-10-19
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    **Hint** $\rm\,\ (2\,y)^3 =\, 8\,y^3\ \, $2012-10-19
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    $(2a^2+2)^3=[2(a^2+1)]^3=2^3(a^2+1)^3$ This is true for any $a$, for $a=\tan x$ you get your equality.2013-08-05

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Since $$2\tan^2 x + 2 = 2(\tan^2 x + 1),$$ we see that $$(2\tan^2 x + 2)^3 = 2^3(\tan^2 x + 1)^3.$$

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    *facepalm. It's so obvious. Thanks.2012-10-19