integral of (6x-9)/(x^2-3x+5)

解

\int \frac{6 x - 9}{x ^{2} - 3 x + 5} d x

- \frac{18}{11} arctan (\frac{1}{11} (2 x - 3)) + \frac{18}{11} arctan (\frac{1}{11} (2 x - 3)) + 3 ln 4 x^{2} - 12 x + 20 + C

解答ステップ

\int \frac{6 x - 9}{x ^{2} - 3 x + 5} d x

部分積分を適用する

= \frac{2}{11} arctan (\frac{1}{11} (2 x - 3)) (6 x - 9) - \int \frac{12}{11} arctan (\frac{1}{11} (2 x - 3)) d x

\int \frac{12}{11} arctan (\frac{1}{11} (2 x - 3)) d x = \frac{12}{11} x arctan (\frac{1}{11} (2 x - 3)) - 3 ln 4 x^{2} - 12 x + 20 - \frac{18}{11} arctan (\frac{2 x - 3}{11})

= \frac{2}{11} arctan (\frac{1}{11} (2 x - 3)) (6 x - 9) - (\frac{12}{11} x arctan (\frac{1}{11} (2 x - 3)) - 3 ln 4 x^{2} - 12 x + 20 - \frac{18}{11} arctan (\frac{2 x - 3}{11}))

簡素化

\frac{2}{11} arctan (\frac{1}{11} (2 x - 3)) (6 x - 9) - (\frac{12}{11} x arctan (\frac{1}{11} (2 x - 3)) - 3 ln 4 x^{2} - 12 x + 20 - \frac{18}{11} arctan (\frac{2 x - 3}{11})) : - \frac{18}{11} arctan (\frac{1}{11} (2 x - 3)) + \frac{18}{11} arctan (\frac{1}{11} (2 x - 3)) + 3 ln 4 x^{2} - 12 x + 20

= - \frac{18}{11} arctan (\frac{1}{11} (2 x - 3)) + \frac{18}{11} arctan (\frac{1}{11} (2 x - 3)) + 3 ln 4 x^{2} - 12 x + 20

定数を解答に追加する

= - \frac{18}{11} arctan (\frac{1}{11} (2 x - 3)) + \frac{18}{11} arctan (\frac{1}{11} (2 x - 3)) + 3 ln 4 x^{2} - 12 x + 20 + C