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juanberrio8a 2022-06-21 Answered

The problem is the integral of

\int \frac{- 8 x}{x^{4} - a^{4}} d x

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humusen6p

Answered 2022-06-22 Author has 22 answers

Noting that

I = \int \frac{- 8 x}{x^{4} - a^{4}} d x = - 4 \int \frac{d (x^{2})}{{(x^{2})}^{2} - {(a^{2})}^{2}}

Let

y = x^{2}

and

b = a^{2}

, then

\begin{aligned} I & = - 4 \int \frac{d y}{y^{2} - b^{2}} \\ = - 4 \int \frac{1}{2 b} (\frac{1}{y - b} - \frac{1}{y + b}) d y \\ = - \frac{2}{b} (\ln | y - b | - \ln | y + b |) + C \\ = - \frac{2}{a^{2}} \ln | \frac{x^{2} - a^{2}}{x^{2} + a^{2}} | + C \\ = \frac{2}{a^{2}} \ln | \frac{x^{2} + a^{2}}{x^{2} - a^{2}} | + C \end{aligned}

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