x-n-1-x-3n-1-x-n-a-dx-

Question Number 144705 by imjagoll last updated on 28/Jun/21

\int \frac{x^{n - 1}}{x^{3 n + 1} (x^{n} - a)} dx ?

Answered by liberty last updated on 28/Jun/21

let x^{n} = y

I = \int \frac{x^{n - 1}}{x^{3 n + 1} (x^{n} - a)} dx

I = \frac{1}{n} \int \frac{dy}{y^{4} (y - a)}

I = \frac{1}{n} \int \frac{1}{a^{4}} (\frac{1}{y - a} - \frac{1}{y} - \frac{a}{y^{2}} - \frac{a^{2}}{y^{3}} - \frac{a^{3}}{y^{4}}) dy

= \frac{1}{{na}^{4}} (\ln ∣ y - a ∣ - \ln ∣ y ∣ + \frac{a}{y} + \frac{a^{2}}{{2 y}^{2}} + \frac{a^{3}}{{3 y}^{3}}) + c

= \frac{1}{{na}^{4}} (\ln ∣ \frac{x^{n} - a}{x^{n}} ∣ + \frac{a}{x^{n}} + \frac{a^{2}}{{2 x}^{2 n}} + \frac{a^{3}}{{3 x}^{3 n}}) + c

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