CBSE 2023 · Region 1 · Set 1 · Q31 · 3 marks
Evaluate : \[\int_{0}^{\frac{\pi}{2}} \mathrm{e}^{\mathrm{x}} \sin \mathrm{x} d \mathrm{x} \]Find : \[\int \frac{1}{\cos (\mathrm{x}-\mathrm{a}) \cos (\mathrm{x}-\mathrm{b})} d \mathrm{x} \]
Evaluate : \[\int_{0}^{\frac{\pi}{2}} \mathrm{e}^{\mathrm{x}} \sin \mathrm{x} d \mathrm{x} \]
Find : \[\int \frac{1}{\cos (\mathrm{x}-\mathrm{a}) \cos (\mathrm{x}-\mathrm{b})} d \mathrm{x} \]
Marking-scheme solution
(a)
Let $\displaystyle \mathrm{I}=\int \mathrm{e}^{\mathrm{x}} \sin \mathrm{x} \mathrm{dx}$
$$=\mathrm{e}^{\mathrm{x}} \sin \mathrm{x}-\int \cos \mathrm{xe}^{\mathrm{x}} \mathrm{dx}
$$OR
(b)Let $\displaystyle \mathrm{I}=\int \frac{1}{\cos (\mathrm{x}-\mathrm{a}) \cos (\mathrm{x}-\mathrm{b})} \mathrm{dx}$
$$=\frac{1}{\sin (\mathrm{a}-\mathrm{b})} \int \frac{\sin [(\mathrm{x}-\mathrm{b})-(\mathrm{x}-\mathrm{a})]}{\cos (\mathrm{x}-\mathrm{a}) \cos (\mathrm{x}-\mathrm{b})} \mathrm{dx}
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CBSE Class 12 Mathematics past-paper question from the 2023board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.