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Question Number 205558 by universe last updated on 24/Mar/24
∫_0 ^(π/2) ((sin^2 4θ )/(sin^2 θ ))dθ = ?
$$\:\:\:\:\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \:\frac{\mathrm{sin}^{\mathrm{2}} \mathrm{4}\theta\:}{\mathrm{sin}^{\mathrm{2}} \theta\:}{d}\theta\:\:=\:\:\:? \\ $$
Answered by Berbere last updated on 24/Mar/24
sin^2 (4x)=4sin^2 (2x)cos^2 (2x) =16sin^2 (x)cos^2 (x)cos^2 (2x) ∫_0 ^(π/2) 8cos^2 (x)cos^2 (2x)=∫_0 ^(π/2) 4(cos(2x)+1)(cos(4x)+1)dx =∫_0 ^(π/2) 4cos(2x)cos(4x)+4cis(2x)+4cos(4x)+4 dx =∫_0 ^(π/2) 6cos(2x)+2cos(6x)+4cos(4x)+4dx =2π
$${sin}^{\mathrm{2}} \left(\mathrm{4}{x}\right)=\mathrm{4}{sin}^{\mathrm{2}} \left(\mathrm{2}{x}\right){cos}^{\mathrm{2}} \left(\mathrm{2}{x}\right) \\ $$$$=\mathrm{16}{sin}^{\mathrm{2}} \left({x}\right){cos}^{\mathrm{2}} \left({x}\right){cos}^{\mathrm{2}} \left(\mathrm{2}{x}\right) \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{8}{cos}^{\mathrm{2}} \left({x}\right){cos}^{\mathrm{2}} \left(\mathrm{2}{x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{4}\left({cos}\left(\mathrm{2}{x}\right)+\mathrm{1}\right)\left({cos}\left(\mathrm{4}{x}\right)+\mathrm{1}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{4}{cos}\left(\mathrm{2}{x}\right){cos}\left(\mathrm{4}{x}\right)+\mathrm{4}{cis}\left(\mathrm{2}{x}\right)+\mathrm{4}{cos}\left(\mathrm{4}{x}\right)+\mathrm{4}\:{dx} \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{6}{cos}\left(\mathrm{2}{x}\right)+\mathrm{2}{cos}\left(\mathrm{6}{x}\right)+\mathrm{4}{cos}\left(\mathrm{4}{x}\right)+\mathrm{4}{dx} \\ $$$$=\mathrm{2}\pi \\ $$$$ \\ $$
Commented by Frix last updated on 24/Mar/24
((sin^2 4x)/(sin^2 x))=2cos 6x +4cos 4x +6cos 2x +4 ⇒ ∫_0 ^(π/2) ((sin^2 4x)/(sin^2 x))dx= =[(1/3)sin 6x +sin 4x +3sin 2x +4x]_0 ^(π/2) =2π
$$\frac{\mathrm{sin}^{\mathrm{2}} \:\mathrm{4}{x}}{\mathrm{sin}^{\mathrm{2}} \:{x}}=\mathrm{2cos}\:\mathrm{6}{x}\:+\mathrm{4cos}\:\mathrm{4}{x}\:+\mathrm{6cos}\:\mathrm{2}{x}\:+\mathrm{4} \\ $$$$\Rightarrow \\ $$$$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\frac{\mathrm{sin}^{\mathrm{2}} \:\mathrm{4}{x}}{\mathrm{sin}^{\mathrm{2}} \:{x}}{dx}= \\ $$$$=\left[\frac{\mathrm{1}}{\mathrm{3}}\mathrm{sin}\:\mathrm{6}{x}\:+\mathrm{sin}\:\mathrm{4}{x}\:+\mathrm{3sin}\:\mathrm{2}{x}\:+\mathrm{4}{x}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} =\mathrm{2}\pi \\ $$

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