Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 96979 by  M±th+et+s last updated on 05/Jun/20

$${\int }_0^{\pi }arctan\left(3^{cos\left(x\right)}\right)dx$$

Answered by MJS last updated on 06/Jun/20

$$y=\arctan 3^{\cos x};x\in [0;\pi ]$$$$\mathrm{this}\mathrm{is}\mathrm{point}\mathrm{symmetric},\mathrm{the}\mathrm{center}\mathrm{is}$$$$C=\left(\begin{array}{c}\frac{\pi }{2}\\ \frac{\pi }{4}\end{array}\right)$$$$\Rightarrow$$$$\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{integral}\mathrm{is}\frac{{\pi }^2}{4}$$

Commented by  M±th+et+s last updated on 06/Jun/20

$$welldonesir$$

Commented by  M±th+et+s last updated on 06/Jun/20

$$sirmjscanyouexplaintheruleand$$$$whencaniuseitandthankyou$$

Commented by MJS last updated on 06/Jun/20

$$\mathrm{we}\mathrm{have}\mathrm{to}\mathrm{find}\mathrm{out}\mathrm{if}{\mathrm{there}}^{\prime }\mathrm{s}\mathrm{some}\mathrm{kind}\mathrm{of}$$$$\mathrm{symmetry}\mathrm{within}\mathrm{the}\mathrm{interval}...{\mathrm{I}}^{\prime }\mathrm{m}\mathrm{not}$$$$\mathrm{sure}\mathrm{if}{\mathrm{there}}^{\prime }\mathrm{s}\mathrm{a}\mathrm{rule}$$$$\mathrm{here}$$$$\arctan \frac{1}{\alpha }=-\arctan \alpha$$$$\alpha =3^{\cos x};-1⩽\cos x⩽1\Rightarrow \frac{1}{3}⩽\alpha ⩽3$$$$\Rightarrow \mathrm{center}\mathrm{is}\mathrm{at}\cos x=0$$$$\mathrm{the}\mathrm{rest}\mathrm{is}\mathrm{easy}$$

Commented by  M±th+et+s last updated on 06/Jun/20

$$thanx$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com