Q63 / 160 · AP EAMCET 2020 (21 Sep Shift 2) · Mathematics

\(\int x^{2020}\left(\tan ^{-1} x+\cot ^{-1} x\right) d x=\)

  • A
    \(\frac{x^{2021}}{2020}\left(\tan ^{-1} x+\cot ^{-1} x\right)+c\)
  • Correct answer
    B
    \(\frac{x^{2021}}{2021}\left(\tan ^{-1} x+\cot ^{-1} x\right)+c\)
  • C
    \(\frac{\pi x^{2021}}{2021}+\frac{\pi}{2}+C\)
  • D
    \(\frac{x^{52}}{52}+\frac{\pi}{2}+C\)

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.

Hence, the correct answer is option: 'B'.

Relevant Questions