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積分問題24番
思考
部分分数分解します。(定石通りですが、大変です。)
計算
\(\displaystyle\int\displaystyle\frac{x-4}{x^4-1} dx \)
\(=-\displaystyle\frac{3}{4}\displaystyle\int\displaystyle\frac{dx}{x-1} +\displaystyle\frac{5}{4}\displaystyle\int\displaystyle\frac{dx}{x+1}\)\(-\displaystyle\frac{1}{2}\displaystyle\int\displaystyle\frac{x-4}{x^2+1}dx\)
\(=-\displaystyle\frac{3}{4}\displaystyle\int\displaystyle\frac{dx}{x-1} +\displaystyle\frac{5}{4}\displaystyle\int\displaystyle\frac{dx}{x+1}\)\(-\displaystyle\frac{1}{4}\displaystyle\int\displaystyle\frac{2x}{x^2+1}dx+\displaystyle\int\displaystyle\frac{2}{x^2+1}dx\)
※最終項の分子を分解
\(=-\displaystyle\frac{3}{4}\log|x-1|+\displaystyle\frac{5}{4}\log|x+1|-\displaystyle\frac{1}{4}\log(x^2+1)+2\tan^{-1} x+C\)
答え
\(-\displaystyle\frac{3}{4}\log|x-1|+\displaystyle\frac{5}{4}\log|x+1|-\displaystyle\frac{1}{4}\log(x^2+1)+2\tan^{-1} x+C\)
