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  4. ∫ sin x ⋅ cos x ⋅ cos 2 x ⋅ cos 4 x ⋅ cos 8 x d x is equal to

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Q. $\int \sin x \cdot \cos x \cdot \cos 2 x \cdot \cos 4 x \cdot \cos 8 x d x$ is equal to

Integrals

Solution:

$I=\int \sin x \cdot \cos x \cdot \cos 2 x \cdot \cos 4 x \cdot \cos 8 x d x$
$=\frac{1}{2} \int \sin 2 x \cdot \cos 2 x \cdot \cos 4 x \cdot \cos 8 x d x$
$=\frac{1}{4} \int \sin 4 x \cdot \cos 4 x \cdot \cos 8 x d x$
$=\frac{1}{8} \int \sin 8 x \cdot \cos 8 x d x$
$=\frac{1}{16} \int \sin 16 x d x=\frac{-1}{256} \cos 16 x+C$