integral of x/((x+3)(x^2-4x+5)^2)

解

\int \frac{x}{( x + 3 ) ( x ^{2} - 4 x + 5 ) ^{2}} d x

- \frac{3}{676} ln ∣ x + 3 ∣ + \frac{1}{1352} (3 ln x^{2} - 4 x + 5 - 30 arctan (x - 2)) + \frac{22 x ^{2} arctan ( x - 2 ) + 5 x ^{2} sin ( 2 arctan ( x - 2 ) ) + 12 x - 88 x arctan ( x - 2 ) - 20 x sin ( 2 arctan ( x - 2 ) ) + 110 arctan ( x - 2 ) + 25 sin ( 2 arctan ( x - 2 ) ) - 30}{104 ( x ^{2} - 4 x + 5 )} + C

解答ステップ

\int \frac{x}{( x + 3 ) ( x ^{2} - 4 x + 5 ) ^{2}} d x

以下の部分分数を得る:

\frac{x}{( x + 3 ) ( x ^{2} - 4 x + 5 ) ^{2}} : - \frac{3}{676 ( x + 3 )} + \frac{3 x - 21}{676 ( x ^{2} - 4 x + 5 )} + \frac{3 x + 5}{26 ( x ^{2} - 4 x + 5 ) ^{2}}

= \int - \frac{3}{676 ( x + 3 )} + \frac{3 x - 21}{676 ( x ^{2} - 4 x + 5 )} + \frac{3 x + 5}{26 ( x ^{2} - 4 x + 5 ) ^{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{3}{676 ( x + 3 )} d x + \int \frac{3 x - 21}{676 ( x ^{2} - 4 x + 5 )} d x + \int \frac{3 x + 5}{26 ( x ^{2} - 4 x + 5 ) ^{2}} d x

\int \frac{3}{676 ( x + 3 )} d x = \frac{3}{676} ln ∣ x + 3 ∣

\int \frac{3 x - 21}{676 ( x ^{2} - 4 x + 5 )} d x = \frac{1}{1352} (3 ln x^{2} - 4 x + 5 - 30 arctan (x - 2))

\int \frac{3 x + 5}{26 ( x ^{2} - 4 x + 5 ) ^{2}} d x = \frac{22 x ^{2} arctan ( x - 2 ) + 5 x ^{2} sin ( 2 arctan ( x - 2 ) ) + 12 x - 88 x arctan ( x - 2 ) - 20 x sin ( 2 arctan ( x - 2 ) ) + 110 arctan ( x - 2 ) + 25 sin ( 2 arctan ( x - 2 ) ) - 30}{104 ( x ^{2} - 4 x + 5 )}

= - \frac{3}{676} ln ∣ x + 3 ∣ + \frac{1}{1352} (3 ln x^{2} - 4 x + 5 - 30 arctan (x - 2)) + \frac{22 x ^{2} arctan ( x - 2 ) + 5 x ^{2} sin ( 2 arctan ( x - 2 ) ) + 12 x - 88 x arctan ( x - 2 ) - 20 x sin ( 2 arctan ( x - 2 ) ) + 110 arctan ( x - 2 ) + 25 sin ( 2 arctan ( x - 2 ) ) - 30}{104 ( x ^{2} - 4 x + 5 )}

定数を解答に追加する

= - \frac{3}{676} ln ∣ x + 3 ∣ + \frac{1}{1352} (3 ln x^{2} - 4 x + 5 - 30 arctan (x - 2)) + \frac{22 x ^{2} arctan ( x - 2 ) + 5 x ^{2} sin ( 2 arctan ( x - 2 ) ) + 12 x - 88 x arctan ( x - 2 ) - 20 x sin ( 2 arctan ( x - 2 ) ) + 110 arctan ( x - 2 ) + 25 sin ( 2 arctan ( x - 2 ) ) - 30}{104 ( x ^{2} - 4 x + 5 )} + C