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Let $f (x) = \frac{1}{1 - \sin^{4} x}$ and $F$ be an antiderivative
of $f$ .
Statement-1: $F (x) = \frac{1}{2} \tan x + \frac{1}{2 \sqrt{2}} \tan^{- 1} (\sqrt{2} \tan x) + C$
Statement-2: $F$ is a one-one function of $\tan x$ .

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STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation forSTATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is False

STATEMENT-1 is False, STATEMENT-2 is True

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detailed solution

Correct option is B

f(x)=1cos2⁡x1+sin2⁡x=sec2⁡x1+tan2⁡x1+2tan2⁡xSo, F(x)=∫1+t21+2t2dt(t=tan⁡x)=∫12+141t2+1/2dt=12t+142tan−1⁡(2tan⁡x)+C=12tan⁡x+122tan−1⁡(2tan⁡x)+Cwhich is a 1 – 1 function of tan x.

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Let f(x)=11−sin4⁡x and F be an antiderivativeof f.Statement-1: F(x)=12tan⁡x+122tan−1⁡(2tan⁡x)+CStatement-2: F is a one-one function of tan x.

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Let $f (x) = \frac{1}{1 - \sin^{4} x}$ and $F$ be an antiderivative
of $f$ .
Statement-1: $F (x) = \frac{1}{2} \tan x + \frac{1}{2 \sqrt{2}} \tan^{- 1} (\sqrt{2} \tan x) + C$
Statement-2: $F$ is a one-one function of $\tan x$ .