Q71 / 125 · BITSAT 2009 · Mathematics

Evaluate: \(\int x \tan ^{-1} x d x\)

  • Correct answer
    A
    \(\frac{1}{2}\left(x^2+1\right) \tan ^{-1} x-\frac{1}{2} x+c\)
  • B
    \(\frac{1}{2}\left(x^2+1\right) \tan ^{-1} x+\frac{1}{2} x+c\)
  • C
    \(\frac{1}{2}\left(\mathrm{x}^2-1\right) \tan ^{-1} \mathrm{x}-\frac{1}{2} \mathrm{x}+\mathrm{c}\)
  • D
    None of these

Solution

\(\begin{aligned} & \int x \tan ^{-1} x d x=\frac{x^2}{2} \tan ^{-1} x-\frac{1}{2} \int \frac{x^2}{1+x^2} d x \\ & =\frac{x^2}{2} \tan ^{-1} x-\frac{1}{2} \int\left[1-\frac{1}{1+x^2}\right] d x \\ & =\frac{1}{2}\left(x^2+1\right) \tan ^{-1} x-\frac{1}{2} x+c \end{aligned}\)

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

More Definite Integration Questions