∫dxsin4x+cos4 x is equal to

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Question

\int \frac{dx}{{sin}^{4} x + {cos}^{4} x}

is equal to

Options:

A . 1√2tan−1(1√2tan 2x)+C

B . √2tan−1(1√2tan 2x)+C

C . 1√2tan−1(1√2cot 2x)+C

D . None of these

Answer: Option A
:
A

\int \frac{dx}{{sin}^{4} x + {cos}^{4} x} = \int \frac{dx}{({sin}^{2} x + {cos}^{2} x)^{2} - 2 {sin}^{2} x {cos}^{2} x} = \int \frac{2 dx}{2 - {sin}^{2} 2 x} = \int \frac{2 {sec}^{2} 2 x}{2 {sec}^{2} 2 x - {tan}^{2} 2 x} dx = 2 \int \frac{{sec}^{2} 2 x}{2 + {tan}^{2} 2 x} dx = \int \frac{dt}{(\sqrt{2})^{2} + t^{2}} [p u t t i n g tan 2 x = t \Rightarrow 2 {sec}^{2} 2 x dx = dt] = \frac{1}{\sqrt{2}} {tan}^{- 1} (\frac{t}{\sqrt{2}}) + C = \frac{1}{\sqrt{2}} {tan}^{- 1} (\frac{1}{\sqrt{2}} tan 2 x) + C

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