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Mathematics · 2022 · 2 marks
CBSE 2022 · Region 1 · Set 2 · Q1
(a) If d/(d x)[ F (x)]=(sec^4 x)/(cosec^4 x) and F ((π)/4)=(π)/4, then find F (x).
(a)
If $\displaystyle \frac{\mathrm{d}}{\mathrm{d} x}[\mathrm{~F}(x)]=\frac{\sec ^{4} x}{\operatorname{cosec}^{4} x}$ and $\displaystyle \mathrm{F}\left(\frac{\pi}{4}\right)=\frac{\pi}{4}$, then find $\displaystyle \mathrm{F}(x)$.
OR
(b)
Find : $\displaystyle \int \frac{\log x-3}{(\log x)^{4}} \mathrm{~d} x$.
Marking-scheme solution
\[\begin{aligned}
F(x) & =\int \tan ^{4} x d x=\int \tan ^{2} x \cdot\left(\sec ^{2} x-1\right) d x \\
& =\int\left(\tan ^{2} x \sec ^{2} x-\sec ^{2} x+1\right) d x \\
F(x) & =\frac{\tan ^{3} x}{3}-\tan x+x+C \\
& x=\frac{\pi}{4}, F(x)=\frac{\pi}{4} \text { gives } C=\frac{2}{3} \\
F(x) & =\frac{\tan ^{3} x}{3}-\tan x+x+\frac{2}{3}
\end{aligned}
\]
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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.