∫x+x23+x16x1+x13dx equals

Put x=u6∴ I=6∫u5+u3+11+u2du=6∫u3du+6∫du1+u2=3u42+6tan−1⁡u+C=3x232+6tan−1⁡x16+C

∫x+x23+x16x1+x13dx equals

Put x=u6∴ I=6∫u5+u3+11+u2du=6∫u3du+6∫du1+u2=3u42+6tan−1⁡u+C=3x232+6tan−1⁡x16+C

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$\int \frac{x + x^{\frac{2}{3}} + x^{\frac{1}{6}}}{x (1 + x^{\frac{1}{3}})} d x equals$

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$\frac{3 x^{\frac{2}{3}}}{2} + 6 \tan^{- 1} (x^{\frac{1}{6}}) + C$

$\frac{3 x^{\frac{2}{3}}}{10} + 6 \tan^{- 1} (x^{\frac{1}{6}}) + C$

$\frac{3 x^{\frac{2}{3}}}{4} + 6 \tan^{- 1} (x^{\frac{1}{6}}) + C$

$\frac{3 x^{\frac{2}{3}}}{5} + 6 \tan^{- 1} (x^{\frac{1}{6}}) + C$

answer is B.

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Detailed Solution

$Put x = u^{6}$

$\begin{array}{l} ∴ I = 6 \int \frac{u^{5} + u^{3} + 1}{1 + u^{2}} du \\ = 6 \int u^{3} du + 6 \int \frac{du}{1 + u^{2}} \\ = \frac{3 u^{4}}{2} + 6 \tan^{- 1} u + C \\ = \frac{3 x^{\frac{2}{3}}}{2} + 6 \tan^{- 1} (x^{\frac{1}{6}}) + C \end{array}$

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$\int \frac{x + x^{\frac{2}{3}} + x^{\frac{1}{6}}}{x (1 + x^{\frac{1}{3}})} d x equals$