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\( \int \frac{x^{2}+a^{2}}{x^{4}+a^{4}} d x \) |
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Answer» \(\int\frac{x^2+a^2}{x^4+a^4}dx\) = \(\int\frac{x^2+a^2}{(x^2+a^2)^2-4a^2x^2}dx\) = \(\int\frac{x^2+a^2}{(x^2+a^2-2ax)(x^2+a^2+2ax)}dx\) (\(\because\) a2 - b2 = (a + b) (a - b)) = \(\int\frac{x^2+a^2}{(x-a)^2(x+a)^2}dx\) = \(\frac12\int(\frac1{(x-a)^2}\frac1{(x+a)^2})dx\) = \(\frac12(-\frac1{x-a}-\frac1{x+a})+c\) = \(-\frac12(\frac1{x+a}+\frac1{x-a})+c\) = \(-\frac12(\frac{2x}{x^2-a^2})+c\) = \(\frac{x}{a^2-x^2}+c\) |
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