Let $$f(x)=2$$−$$sin$$⁡$$x2+$$$$cos$$⁡$$xStatement-1$$: The antiderivative of f is a periodic functionof period π.$$Statement-2$$: The period of the $$functiong(x)=log$$⁡$$(2+$$$$cos$$⁡$$x)+Atan$$−$$1$$⁡$$(Btan$$⁡$$x/2)+C$$,A,B,C being contants is $$2$$π.

Question

Let $$f(x)=2$$−$$sin$$⁡$$x2+$$$$cos$$⁡$$xStatement-1$$: The antiderivative of f is a periodic functionof period π.$$Statement-2$$: The period of the $$functiong(x)=log$$⁡$$(2+$$$$cos$$⁡$$x)+Atan$$−$$1$$⁡$$(Btan$$⁡$$x/2)+C$$,A,B,C being contants is $$2$$π.

Infinity Learn · Accepted Answer

$$F(x)=2I1+I2$$,$$I1=$$∫$$dx2+$$$$cos$$⁡x and $$I2=$$−∫$$sin$$⁡$$x2+$$$$cos$$⁡$$xdx=log$$⁡$$(2+$$$$cos$$⁡$$x)+$$ constant. $$I1=$$∫$$2dt1+t22+1$$−$$t21+t2=2$$∫$$dt3+t2=23tan$$−$$1$$⁡$$13tan$$⁡$$(x/2)+C$$ $$(t=$$$$tan$$⁡$$(x/2))Hence$$ $$F(x)=log$$⁡$$(2+$$$$cos$$⁡$$x)+43tan$$−$$1$$⁡$$13tan$$⁡$$(x/2)+C$$ which is periodic with period $$2$$π.

Let $f (x) = \frac{2 - \sin x}{2 + \cos x}$
Statement-1: The antiderivative of f is a periodic function
of period $π$ .
Statement-2: The period of the function
$g (x) = \log (2 + \cos x) + A \tan^{- 1} (B \tan x / 2) + C, A, B, C$ being contants is $2 π$ .

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Let f(x)=2−sin⁡x2+cos⁡xStatement-1: The antiderivative of f is a periodic functionof period π.Statement-2: The period of the functiong(x)=log⁡(2+cos⁡x)+Atan−1⁡(Btan⁡x/2)+C,A,B,C being contants is 2π.

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Ready to Test Your Skills?

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Let $f (x) = \frac{2 - \sin x}{2 + \cos x}$
Statement-1: The antiderivative of f is a periodic function
of period $π$ .
Statement-2: The period of the function
$g (x) = \log (2 + \cos x) + A \tan^{- 1} (B \tan x / 2) + C, A, B, C$ being contants is $2 π$ .