\(\int x^{2020}\left(\tan ^{-1} x+\cot ^{-1} x\right) d x=\)
\(\int x^{2020}\left(\tan ^{-1} x+\cot ^{-1} x\right) d x=\)
- \(\frac{x^{2021}}{2020}\left(\tan ^{-1} x+\cot ^{-1} x\right)+c\)
- \(\frac{x^{2021}}{2021}\left(\tan ^{-1} x+\cot ^{-1} x\right)+c\)
- \(\frac{\pi x^{2021}}{2021}+\frac{\pi}{2}+C\)
- \(\frac{x^{52}}{52}+\frac{\pi}{2}+C\)
Solution
\(\begin{aligned}
I & =\int x^{2020}\left(\tan ^{-1} x+\cot ^{-1} x\right) d x \\
& =\int x^{2020}\left(\frac{\pi}{2}\right) d x \quad\left\{\because \tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}\right\} \\
& =\frac{\pi}{2} \frac{x^{2021}}{2021}+C=\frac{x^{2021}}{2021}\left(\tan ^{-1} x+\cot ^{-1} x\right)+C
\end{aligned}\)
Hence, option (b) is correct.
Asked in: AP EAMCET 2020 (21 Sep Shift 2)
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