If ∫1x(x5−1)(x5+1)dx=Alogx+Blog(x5−1)+Clog(x5+1) then (A,B,C)=

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If $\int \frac{1}{x (x^{5} - 1) (x^{5} + 1)} d x = A \log x + B \log (x^{5} - 1) + C \log (x^{5} + 1)$ then $(A, B, C) =$

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(- 1, \frac{1}{10}, \frac{1}{10})

(- \frac{5}{4}, \frac{1}{8}, - \frac{1}{8})

(- \frac{5}{4}, - \frac{1}{8}, \frac{1}{8})

(1, \frac{1}{10}, - \frac{1}{10})

answer is A.

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\int \frac{1}{x (x^{5} - 1) (x^{5} + 1)} d x = A \log x + B \log (x^{5} - 1) + C \log (x^{5} + 1)

$= \int \frac{x^{4} . d x}{x^{5} (x^{5} - 1) (x^{5} + 1)}$

Put $x^{5} = t$

$5 x^{4} d x = d t$

$\Rightarrow \frac{1}{5} \int \frac{d t}{t (t - 1) (t + 1)} = \frac{1}{5} \int (\frac{P}{t} + \frac{Q}{t - 1} + \frac{R}{t + 1}) d t$

$= \frac{P}{5} . \log t + \frac{Q}{5} . \log | t - 1 | + \frac{R}{5} . \log | t + 1 |$

$p = - 1, Q = \frac{1}{2}, R = \frac{1}{2}$

$∴ \frac{p}{5} = \frac{- 1}{5}; B = \frac{Q}{5} = \frac{1}{10}, C = \frac{R}{5} = \frac{1}{10}$

$∴ (A, B, C) = (- 1, \frac{1}{10}, \frac{1}{10})$

$= \frac{1}{5} \log x^{5} + \frac{1}{10} \log | x^{5} - 1 | + \frac{1}{10} \log | x^{5} + 1 | + C$

$= \frac{- 1}{5} \times 5 \log x + \frac{1}{10} \log | x^{5} - 1 | + \frac{1}{10} \log (x^{5} + 1) + C$

$∴ (A, B, C) = (- 1, \frac{1}{10}, \frac{1}{10})$

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