Find the integrals of the functions in Exercises: ^42x - Vidyadip

Question

Find the integrals of the functions in Exercises:
$\cos^42\text{x}$

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Answer

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

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