integral of (x^2-3x+12)/((x^2-4x+11)^2)

解

\int \frac{x ^{2} - 3 x + 12}{( x ^{2} - 4 x + 11 ) ^{2}} d x

\frac{1}{14 7} x arctan (\frac{1}{7} (x - 2)) + \frac{1}{28 7} x sin (2 arctan (\frac{1}{7} (x - 2))) + \frac{15}{14 7} arctan (\frac{1}{7} (x - 2)) + \frac{1}{28 7} sin (2 arctan (\frac{1}{7} (x - 2))) - \frac{( x - 2 ) ( arctan ( \frac{1}{7} ( x - 2 ) ) ( x ^{2} - 4 x + 11 ) + 7 ( x - 2 ) )}{14 7 ( x ^{2} - 4 x + 11 )} - \frac{1}{2 ( x ^{2} - 4 x + 11 )} + C

解答ステップ

\int \frac{x ^{2} - 3 x + 12}{( x ^{2} - 4 x + 11 ) ^{2}} d x

以下の部分分数を得る:

\frac{x ^{2} - 3 x + 12}{( x ^{2} - 4 x + 11 ) ^{2}} : \frac{1}{x ^{2} - 4 x + 11} + \frac{x + 1}{( x ^{2} - 4 x + 11 ) ^{2}}

= \int \frac{1}{x ^{2} - 4 x + 11} + \frac{x + 1}{( x ^{2} - 4 x + 11 ) ^{2}} d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{x ^{2} - 4 x + 11} d x + \int \frac{x + 1}{( x ^{2} - 4 x + 11 ) ^{2}} d x

\int \frac{1}{x ^{2} - 4 x + 11} d x = \frac{1}{7} arctan (\frac{x - 2}{7})

\int \frac{x + 1}{( x ^{2} - 4 x + 11 ) ^{2}} d x = \frac{1}{28 7} (2 arctan (\frac{x - 2}{7}) + sin (2 arctan (\frac{x - 2}{7}))) (x + 1) - \frac{1}{14} \frac{( x - 2 ) ( arctan ( \frac{1}{7} ( x - 2 ) ) ( x ^{2} - 4 x + 11 ) + 7 ( x - 2 ) )}{7 ( x ^{2} - 4 x + 11 )} + \frac{7}{x ^{2} - 4 x + 11}

= \frac{1}{7} arctan (\frac{x - 2}{7}) + \frac{1}{28 7} (2 arctan (\frac{x - 2}{7}) + sin (2 arctan (\frac{x - 2}{7}))) (x + 1) - \frac{1}{14} \frac{( x - 2 ) ( arctan ( \frac{1}{7} ( x - 2 ) ) ( x ^{2} - 4 x + 11 ) + 7 ( x - 2 ) )}{7 ( x ^{2} - 4 x + 11 )} + \frac{7}{x ^{2} - 4 x + 11}

簡素化

\frac{1}{7} arctan (\frac{x - 2}{7}) + \frac{1}{28 7} (2 arctan (\frac{x - 2}{7}) + sin (2 arctan (\frac{x - 2}{7}))) (x + 1) - \frac{1}{14} \frac{( x - 2 ) ( arctan ( \frac{1}{7} ( x - 2 ) ) ( x ^{2} - 4 x + 11 ) + 7 ( x - 2 ) )}{7 ( x ^{2} - 4 x + 11 )} + \frac{7}{x ^{2} - 4 x + 11} : \frac{1}{14 7} x arctan (\frac{1}{7} (x - 2)) + \frac{1}{28 7} x sin (2 arctan (\frac{1}{7} (x - 2))) + \frac{15}{14 7} arctan (\frac{1}{7} (x - 2)) + \frac{1}{28 7} sin (2 arctan (\frac{1}{7} (x - 2))) - \frac{( x - 2 ) ( arctan ( \frac{1}{7} ( x - 2 ) ) ( x ^{2} - 4 x + 11 ) + 7 ( x - 2 ) )}{14 7 ( x ^{2} - 4 x + 11 )} - \frac{1}{2 ( x ^{2} - 4 x + 11 )}

= \frac{1}{14 7} x arctan (\frac{1}{7} (x - 2)) + \frac{1}{28 7} x sin (2 arctan (\frac{1}{7} (x - 2))) + \frac{15}{14 7} arctan (\frac{1}{7} (x - 2)) + \frac{1}{28 7} sin (2 arctan (\frac{1}{7} (x - 2))) - \frac{( x - 2 ) ( arctan ( \frac{1}{7} ( x - 2 ) ) ( x ^{2} - 4 x + 11 ) + 7 ( x - 2 ) )}{14 7 ( x ^{2} - 4 x + 11 )} - \frac{1}{2 ( x ^{2} - 4 x + 11 )}

定数を解答に追加する

= \frac{1}{14 7} x arctan (\frac{1}{7} (x - 2)) + \frac{1}{28 7} x sin (2 arctan (\frac{1}{7} (x - 2))) + \frac{15}{14 7} arctan (\frac{1}{7} (x - 2)) + \frac{1}{28 7} sin (2 arctan (\frac{1}{7} (x - 2))) - \frac{( x - 2 ) ( arctan ( \frac{1}{7} ( x - 2 ) ) ( x ^{2} - 4 x + 11 ) + 7 ( x - 2 ) )}{14 7 ( x ^{2} - 4 x + 11 )} - \frac{1}{2 ( x ^{2} - 4 x + 11 )} + C