int_{-3}^{3} cot^{-1} x , dx = | vedclass

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(C) Let $ I = \int_{-3}^{3} \cot^{-1} x \, dx $. Using the property $ \int_{-a}^{a} f(x) \, dx = \int_{-a}^{a} f(-x) \, dx $ is not directly applicabl

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$ 3\pi $

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(C) Let $ I = \int_{-3}^{3} \cot^{-1} x \, dx $. Using the property $ \int_{-a}^{a} f(x) \, dx = \int_{-a}^{a} f(-x) \, dx $ is not directly applicable here,but we know that $ \cot^{-1}(-x) = \pi - \cot^{-1} x $. Consider the integral $ I = \int_{-3}^{3} \cot^{-1} x \, dx $. Using the property $ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a+b-x) \, dx $,we get: $ I = \int_{-3}^{3} \cot^{-1}(-x) \, dx = \int_{-3}^{3} (\pi - \cot^{-1} x) \, dx $. $ I = \int_{-3}^{3} \pi \, dx - \int_{-3}^{3} \cot^{-1} x \, dx $. $ I = \pi [x]_{-3}^{3} - I $. $ 2I = \pi (3 - (-3)) $. $ 2I = 6\pi $. $ I = 3\pi $.

$ \int_{-3}^{3} \cot^{-1} x \, dx = $

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