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If $\int \frac{d x}{\sin^{4} x + \cos^{4} x} = \frac{1}{\sqrt{2}} \tan^{- 1} f (x) + C$ then

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a

f(x)=tan⁡x−cot⁡x

b

f(π/4)=0

c

f (x) is continuous on R

d

f(x)=12(tan⁡x−cot⁡x)

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detailed solution

Correct option is B

Dividing the numerator and denominator by cos4⁡x, the given integral can be written as ∫sec2⁡x1+tan2⁡xtan4⁡x+1dx=∫1+t21+t4dt(t=tan⁡x)=∫1+1/t2t2+1/t2dt=∫1+1/t2t−1t2+2dt=12tan−1⁡12t−1t+C=12tan−1⁡tan⁡x−cot⁡x2+CThus f(x)=tan⁡x−cot⁡x2 so fπ4=0

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