Question

Integrate sin²2x cos²2x dx

Answer 1

$\sin^2x=\dfrac{1-\cos2x}2$
$\cos^2x=\dfrac{1+\cos2x}2$

From the identities above, you have

$\sin^22x\cos^22x=\dfrac{(1-\cos4x)(1+\cos4x)}4=\dfrac{1-\cos^24x}4$

Applying once more, you have

$\dfrac{1-\cos^24x}4=\dfrac{1-\dfrac{1+\cos8x}2}4=\dfrac{1-\cos8x}8$

So,

$\displaystyle\int\sin^22x\cos^22x\,\mathrm dx=\frac18\int(1-\cos8x)\,\mathrm dx=\frac x8-\frac1{64}\sin8x+C$