CBSE 2025 · Region 7 · Set 2 · Q32 · 5 marks
Find : \[\int \frac{x}{(x-1)\left(x^{2}+4\right)} d x \]Evaluate : \[\int_{0}^{\pi} \frac{x \sin x}{1+\cos ^{2} x} d x \]
Find : \[\int \frac{x}{(x-1)\left(x^{2}+4\right)} d x \]
Evaluate : \[\int_{0}^{\pi} \frac{x \sin x}{1+\cos ^{2} x} d x \]
Marking-scheme solution
(a)
$\displaystyle \int \frac{x}{(x-1)\left(x^{2}+4\right)} d x=\frac{1}{5} \int \frac{1}{x-1} d x-\frac{1}{10} \int \frac{2 x}{x^{2}+4} d x+\frac{4}{5} \int \frac{1}{x^{2}+4} d x$, (By Partial Fractions)
\[=\frac{1}{5} \log |x-1|-\frac{1}{10} \log \left(x^{2}+4\right)+\frac{2}{5} \tan ^{-1} \frac{x}{2}+C
\]
IntegralsIntegration by Partial FractionsApplylong_answermedium
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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.