Q66 / 180 · KCET 2019 · Mathematics

\( \int_{-3}^{3} \cot ^{-1} x d x= \)

  • A
    \( 00 \)
  • B
    \( 03 \)
  • Correct answer
    C
    \(3 \pi\)
  • D
    (1)

Solution

(C) \( \int_{-3}^{3} \cot ^{-1} x d x=\int_{-3}^{3} \cot ^{-1} x \cdot 1 d x \) \( =\left[\cot ^{-1} x \cdot x\right]_{-3}^{3}-\int_{-3}^{3} x\left(-\frac{1}{1+x 2}\right) d x \) \( =3 \cot ^{-1} 3-(-3) \cot ^{-1}(-3)+\int_{-3}^{3} \frac{x}{1+x^{2}} d x \) \( =3 \cot ^{-1} 3+3\left(\Pi-\cot ^{-1} 3\right)+\frac{1}{2}\left[\log \left[1+x^{2}\right]\right]_{-3}{ }^{3} \) \( =3 \cot ^{-1} 3+3 \pi-3 \cot ^{-1} 3+\frac{1}{2}[\log 10-\log 10] \) \( =3 \pi \)

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

More Definite Integration Questions