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Prove-That-I-0-sin-x-cos-x-4-x-x-dx-1-32-2-2-1-pi-m-n-




Question Number 150749 by mnjuly1970 last updated on 15/Aug/21
    Prove   That ::          I := ∫_0 ^( ∞)  ((( sin (x ).cos (x ))^( 4) )/(x . (√x) ))dx=(1/(32)) (2 (√2) −1 )(√( π)) ....■       ..m.n..
$$ \\ $$$$\:\:\mathrm{Prove}\:\:\:\mathrm{That}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\mathcal{I}\::=\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\left(\:{sin}\:\left({x}\:\right).{cos}\:\left({x}\:\right)\right)^{\:\mathrm{4}} }{{x}\:.\:\sqrt{{x}}\:}{dx}=\frac{\mathrm{1}}{\mathrm{32}}\:\left(\mathrm{2}\:\sqrt{\mathrm{2}}\:−\mathrm{1}\:\right)\sqrt{\:\pi}\:….\blacksquare\:\: \\ $$$$\:\:\:..{m}.{n}..\:\: \\ $$

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