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Mathematics · 2022 · 3 marks
CBSE 2022 · Region 5 · Set 2 · Q10
Evaluate: ∫_0^π / 2 1/(1+( tan x)^2 / 3) d x
Evaluate : \[\int_{0}^{\pi / 2} \frac{1}{1+(\tan x)^{2 / 3}} d x \]
Marking-scheme solution
\[\begin{aligned}
I & =\int_{0}^{\pi / 2} \frac{d x}{1+(\tan x)^{2 / 3}} \\
& =\int_{0}^{\pi / 2} \frac{\cos ^{2 / 3} x}{\cos ^{2 / 3} x+\sin ^{2 / 3} x} d x \\
& =\int_{0}^{\pi / 2} \frac{\cos ^{2 / 3}\left(\dfrac{\pi}{2}-x\right)}{\cos ^{2 / 3}\left(\dfrac{\pi}{2}-x\right)+\sin ^{2 / 3}\left(\dfrac{\pi}{2}-x\right)} d x \\
& =\int_{0}^{\pi / 2} \frac{\sin ^{2 / 3} x}{\sin ^{2 / 3} x+\cos ^{2 / 3} x} d x \\
& (\text { I })+(\text { II }) \\
& \Rightarrow 2 I=\int_{0}^{\pi / 2} 1 d x=\frac{\pi}{2} \Rightarrow I=\frac{\pi}{4}
\end{aligned}
\]
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CBSE Class 12 Mathematics past-paper question from the 2022 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.