integral of (x-2)/((x^2+8x+64)^2)

解

\int \frac{x - 2}{( x ^{2} + 8 x + 64 ) ^{2}} d x

\frac{1}{768 3} (2 arctan (\frac{1}{4 3} x + \frac{1}{3}) + sin (2 arctan (\frac{1}{4 3} x + \frac{1}{3}))) (x - 2) - \frac{1}{384 3} arctan (\frac{x + 4}{4 3}) (x + 4) + C

解答ステップ

\int \frac{x - 2}{( x ^{2} + 8 x + 64 ) ^{2}} d x

部分積分を適用する

= \frac{1}{768 3} (2 arctan (\frac{1}{4 3} x + \frac{1}{3}) + sin (2 arctan (\frac{1}{4 3} x + \frac{1}{3}))) (x - 2) - \int \frac{1}{768 3} (2 arctan (\frac{1}{4 3} x + \frac{1}{3}) + sin (2 arctan (\frac{1}{4 3} x + \frac{1}{3}))) d x

\int \frac{1}{768 3} (2 arctan (\frac{1}{4 3} x + \frac{1}{3}) + sin (2 arctan (\frac{1}{4 3} x + \frac{1}{3}))) d x = \frac{1}{384 3} arctan (\frac{x + 4}{4 3}) (x + 4)

= \frac{1}{768 3} (2 arctan (\frac{1}{4 3} x + \frac{1}{3}) + sin (2 arctan (\frac{1}{4 3} x + \frac{1}{3}))) (x - 2) - \frac{1}{384 3} arctan (\frac{x + 4}{4 3}) (x + 4)

定数を解答に追加する

= \frac{1}{768 3} (2 arctan (\frac{1}{4 3} x + \frac{1}{3}) + sin (2 arctan (\frac{1}{4 3} x + \frac{1}{3}))) (x - 2) - \frac{1}{384 3} arctan (\frac{x + 4}{4 3}) (x + 4) + C