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Question Number 56200 by naka3546 last updated on 11/Mar/19

Commented by Abdo msup. last updated on 12/Mar/19

$$changementx=sin\theta give$$$$I={\int }_0^{\frac{\pi }{2}}\frac{arcos\left(cos\theta \right)cos\theta }{cos\theta }d\theta ={\int }_0^{\frac{\pi }{2}}\theta d\theta$$$$={\left[\frac{{\theta }^2}{2}\right]}_0^{\frac{\pi }{2}}=\frac{{\pi }^2}{8}.$$

Commented by naka3546 last updated on 12/Mar/19

Commented by naka3546 last updated on 12/Mar/19

$$howaboutthis?Isitsameasbefore?$$

Commented by MJS last updated on 12/Mar/19

$$\mathrm{usually}{\cos }^{n}x\mathrm{means}{\left(\cos x\right)}^n.\mathrm{somehow}$$$${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{only}\mathrm{used}\mathrm{witb}n⩾1.{\cos }^{-1}x\mathrm{means}$$$$\arccos x.\mathrm{this}\mathrm{is}\mathrm{the}\mathrm{same}\mathrm{principle}\mathrm{as}$$$$y=f\left(x\right),x=f^{-1}\left(y\right).\mathrm{but}\mathrm{here}\mathrm{many}\mathrm{people}$$$$\mathrm{write}{f}^{\prime }\left(x\right)\mathrm{for}\mathrm{the}{1}^{\mathrm{st}}\mathrm{derivate}\mathrm{and}{f}^n\left(x\right)$$$$\mathrm{for}\mathrm{the}{n}^{\mathrm{th}}\mathrm{derivate}...\mathrm{once}\mathrm{there}\mathrm{had}\mathrm{been}$$$$\mathrm{rules}\mathrm{but}\mathrm{with}\mathrm{the}\mathrm{internet}\mathrm{confusion}\mathrm{came}...$$

Commented by Kunal12588 last updated on 12/Mar/19

$$iwillgladtoknowthoserules$$$$Iheardthisfromourteacher-$$$$Cosine\left(anytrigfunc\right)shouldstartwith$$$$lowercaseletteronly{}^{\prime }cos{x}^{\prime }isrightbut{}^{\prime }{Cos}^{\prime }$$$$iswrong.nowadaysinternet{doesn}^{\prime }tfollow$$$$thisrule.Alsohesays\sqrt{3}was1.732onhis$$$$daysthen1.73innextgenerationinour$$$$thisis1.7thenournextwouldsay\sqrt{3}=1$$

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