^42x dx - Vidyadip

Question

$\int\cos^42\text{x dx}$

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Answer

$\int\cos^42\text{x dx}$
$=\int(\cos^22\text{x})^2\text{dx}$
$=\int\Big(\frac{1+\cos4\text{x}}{2}\Big)^2\text{dx}$ $\Big[\therefore\cos^2\text{x}=\frac{1+\cos2\text{x}}{2}\Big]$
$=\frac{1}{4}\int(1+\cos4\text{x})^2\text{dx}$
$=\frac{1}{4}\int(1+\cos^24\text{x}+2\cos4\text{x})\text{dx}$
$=\frac{1}{4}\Big[1+\Big(\frac{1+\cos8\text{x}}{2}\Big)+2\cos4\text{x}\Big]\text{dx}$
$=\frac{1}{4}\int\Big(\frac{3}{2}+\frac{\cos8\text{x}}{2}+2\cos4\text{x}\Big]\text{dx}$
$=\frac{1}{4}\Big[\frac{3\text{x}}{2}+\frac{\sin8\text{x}}{16}+\frac{2\sin4\text{x}}{4}\Big]+\text{C}$
$=\frac{3\text{x}}{8}+\frac{\sin8\text{x}}{64}+\frac{\sin4\text{x}}{8}+\text{C}$

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