Question:

Substitute $\tan x = t^2 \Rightarrow \int \dfrac{2t^2}{t^4+1} dt = \int \dfrac{t^2+1}{t^4+1} dt + \int \dfrac{t^2-1}{t^4+1} dt$. [2.0 Marks]
Evaluate $I_1 = \dfrac{1}{\sqrt{2}} \tan^{-1}\left( \dfrac{t^2-1}{\sqrt{2}t} \right)$, $I_2 = \dfrac{1}{2\sqrt{2}} \log\left| \dfrac{t^2 - \sqrt{2}t + 1}{t^2 + \sqrt{2}t + 1} \right|$. [2.0 Marks]
Substitute $t = \sqrt{\tan x}$ for final answer. [1.0 Mark]

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🎯 Official CBSE Marking Scheme:
Substituting $t = \sqrt{\tan x}$ and splitting into twin integrals: 2.0 Marks
Evaluating both $I_1$ and $I_2$ integrals: 2.0 Marks
Re-substituting $t = \sqrt{\tan x}$: 1.0 Mark