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Definite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration2 Marks
MCQ
$\int_{-3}^3 \cot ^{-1} x d x=$
  • A
    $0$
  • B
    3
  • ✓
    $3 \pi$
  • D
    $6 \pi$
✓

Answer

Correct option: C.
$3 \pi$
(C)
$\int_{-3}^3 \cot ^{-1} x d x$
$=\left[x \cdot \cot ^{-1} x\right]_{-3}^3-\int_{-3}^3 x\left(\frac{-1}{1+x^2}\right) d x$
$=(3) \cot ^{-1} 3-(-3) \cot ^{-1}(-3)+\frac{1}{2} \int_{-3}^3 \frac{2 x}{1+x^2} d x$
$=3 \cot ^{-1} 3+3\left(\pi-\cot ^{-1} 3\right)+\frac{1}{2}\left[\log \left(1+x^2\right)\right]_{-3}^3$
$\begin{array}{l}=3 \cot ^{-1} 3+3 \pi-3 \cot ^{-1} 3+\frac{1}{2}[\log 10-\log 10] \\ =3 \pi\end{array}$

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