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The integral \(\int \frac{\left(x^{8}-x^{2}\right) \mathrm{d} x}{\left(x^{12}+3 x^{6}+1\right) \tan ^{-1}\left(x^{3}+\fr…

Q1

The integral \(\int \frac{\left(x^{8}-x^{2}\right) \mathrm{d} x}{\left(x^{12}+3 x^{6}+1\right) \tan ^{-1}\left(x^{3}+\frac{1}{x^{3}}\right)}\) is equal to :

[JEE Main 2024, 27 Jan (Shift 2)]

a

\(\log _{\mathrm{e}}\left(\left|\tan ^{-1}\left(x^{3}+\frac{1}{x^{3}}\right)\right|\right)^{1 / 3}+\mathrm{C}\)

b

\(\log _{\mathrm{e}}\left(\left|\tan ^{-1}\left(x^{3}+\frac{1}{x^{3}}\right)\right|\right)^{3}+\mathrm{C}\)

c

\(\log _{\mathrm{e}}\left(\left|\tan ^{-1}\left(x^{3}+\frac{1}{x^{3}}\right)\right|\right)^{1 / 2}+\mathrm{C}\)

d

\(\log _{\mathrm{e}}\left(\left|\tan ^{-1}\left(x^{3}+\frac{1}{x^{3}}\right)\right|\right)+\mathrm{C}\)

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