=∫1a4-x4dx=∫1a2+x2a2-x2dx =∫12a21a2+x2+1a2-x2dx=12a2∫1a2+x2dx+12a2∫1a2-x2dx =12a2·1aTan-1xa+12a2·12aloga+xa-x+c=14a3loga+xa-x+2Tan-1xa+c

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$\int \frac{1}{a^{4} - x^{4}} dx =$

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$\frac{1}{2 a^{3}} [\log |\frac{a + x}{a - x}| + 2 {Tan}^{- 1} \frac{x}{a}] + c$

$\frac{1}{4 a^{3}} [\log |\frac{a + x}{a - x}| + 2 {Tan}^{- 1} \frac{x}{a}] + c$

$\frac{1}{4 a^{3}} [\log |\frac{a - x}{a + x}| - 2 {Tan}^{- 1} \frac{x}{a}] + c$

answer is A.

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Detailed Solution

$= \int \frac{1}{a^{4} - x^{4}} dx = \int \frac{1}{(a^{2} + x^{2}) (a^{2} - x^{2})} dx = \int \frac{1}{2 a^{2}} [\frac{1}{a^{2} + x^{2}} + \frac{1}{a^{2} - x^{2}}] dx$

$= \frac{1}{2 a^{2}} \int \frac{1}{a^{2} + x^{2}} dx + \frac{1}{2 a^{2}} \int \frac{1}{a^{2} - x^{2}} dx = \frac{1}{2 a^{2}} \cdot \frac{1}{a} {Tan}^{- 1} (\frac{x}{a}) + \frac{1}{2 a^{2}} \cdot \frac{1}{2 a} \log |\frac{a + x}{a - x}| + c$