CBSE 2025 · Region 7 · Set 1 · Q32 · 5 marks
Find : \[\int \frac{\mathrm{x}^{2}+1}{\left(\mathrm{x}^{2}+2\right)\left(2 \mathrm{x}^{2}+1\right)} d \mathrm{x} \]Evaluate : \[\int_{0}^{\pi} \frac{\mathrm{x} \tan \mathrm{x}}{\sec \mathrm{x}+\tan \mathrm{x}} d \mathrm{x} \]
Find : \[\int \frac{\mathrm{x}^{2}+1}{\left(\mathrm{x}^{2}+2\right)\left(2 \mathrm{x}^{2}+1\right)} d \mathrm{x} \]
Evaluate : \[\int_{0}^{\pi} \frac{\mathrm{x} \tan \mathrm{x}}{\sec \mathrm{x}+\tan \mathrm{x}} d \mathrm{x} \]
Marking-scheme solution
$\displaystyle \int \frac{\mathrm{x}^{2}+1}{\left(\mathrm{x}^{2}+2\right)\left(2 \mathrm{x}^{2}+1\right)} d \mathrm{x}=\frac{1}{3} \int \frac{1}{\mathrm{x}^{2}+2} d \mathrm{x}+\frac{1}{3} \int \frac{1}{2 \mathrm{x}^{2}+1} d \mathrm{x} \quad$ (Using Partial Fractions)
\[=\frac{1}{3} \cdot \frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{\mathrm{x}}{\sqrt{2}}\right)+\frac{1}{3} \cdot \frac{1}{\sqrt{2}} \tan ^{-1}(\sqrt{2} \mathrm{x})+C
\]
Let $\displaystyle \mathrm{I}=\int_{0}^{\pi} \frac{\mathrm{x} \boldsymbol{\operatorname { t a n }} \mathrm{x}}{\sec \mathrm{x}+\tan \mathrm{x}} \mathrm{dx}$
\[\Rightarrow \mathrm{I}=\int_{0}^{\pi} \frac{(\pi-\mathrm{x}) \tan \mathrm{x}}{\sec \mathrm{x}+\tan \mathrm{x}} d \mathrm{x}
\]
IntegralsIntegration by Partial FractionsApplylong_answerhard
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CBSE Class 12 Mathematics past-paper question from the 2025board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.