Question

\[\int\limits_0^1 \frac{2x}{1 + x^4} dx\]

Answer 1

\[Let\ I = \int_0^1 \frac{2x}{1 + x^4} d x . \]
\[Let\ x^2 = t . Then, 2x\ dx\ = dt\]
\[When\ x = 0, t = 0\ and\ x\ = 1, t = 1\]
\[ \therefore I = \int_0^1 \frac{2x}{1 + x^4} d x\]
\[ \Rightarrow I = \int_0^1 \frac{1}{1 + t^2} d t\]
\[ \Rightarrow I = \left[ \tan^{- 1} t \right]_0^1 \]
\[ \Rightarrow I = \tan^{- 1} 1 - \tan^{- 1} 0\]
\[ \Rightarrow I = \frac{\pi}{4}\]
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