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$\int_{0}^{2 π} \frac{1}{1 + \tan^{4} x} d x is equal to$

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a

π

b

π2

c

2π

d

3π

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detailed solution

Correct option is A

I=∫02π 11+tan4⁡xdx⇒I=2∫0π dx1+tan4⁡x apply ∫02af(x)dx=2∫0a f(x)dx if f(2a-x)=f(x)⇒I=4∫0π/2 dx1+tan4⁡x…⋯⋯(1) same formula Or I=4∫0π/2 dx1+cot4⁡x⋯(2) apply ∫0af(x)dx=∫0a f(a-x)dxAdding (1) and (2), we get 2I=4∫0π/2 dx=4×π2⇒I=π

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