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Integrate $(\cos x)^4$. I see solutions using power reduction everywhere. I vaguely remember doing it based on some manipulation of trig identities $(\cos x)^2 = 1 - (\sin x)^2$ and $u$-substitution alone.

Anybody know what I am talking about?

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    Just for future reference, this is often written as $\cos^4 x$ just to reduce the noise of parentheses.2012-04-28

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Using basic trigonometric identities, we have

$\begin{aligned}\cos^4(x)&=\cos^2(x)(1-\sin^2(x))\\ &=\cos^2(x)-\sin^2(x)\cos^2(x)\\ &=\cos^2(x)-\dfrac{\sin^2(2x)}{4}\\ &=\dfrac{1+\cos(2x)}{2}-\dfrac{1-\cos(4x)}{8},\end{aligned}$

which should be much more manageable.

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    Alternatively: \begin{align*}\cos^4 x&=\left(\frac{1+\cos\,2x}{2}\right)^2\\&=\frac14+\frac{\cos\,2x}{2}+\frac{\cos^2 2x}{4}\\&=\frac14+\frac{\cos\,2x}{2}+\frac14\frac{1+\cos\,4x}{2}\end{align*}2012-04-29