integral of (6x^2-15x+22)/((x+3)(x^2+2)^2)

解

\int \frac{6 x ^{2} - 15 x + 22}{( x + 3 ) ( x ^{2} + 2 ) ^{2}} d x

ln ∣ x + 3 ∣ - \frac{1}{2} ln x^{2} + 2 + \frac{3}{2} arctan (\frac{x}{2}) + \frac{5}{2 ( x ^{2} + 2 )} + C

解答ステップ

\int \frac{6 x ^{2} - 15 x + 22}{( x + 3 ) ( x ^{2} + 2 ) ^{2}} d x

以下の部分分数を得る:

\frac{6 x ^{2} - 15 x + 22}{( x + 3 ) ( x ^{2} + 2 ) ^{2}} : \frac{1}{x + 3} + \frac{- x + 3}{x ^{2} + 2} - \frac{5 x}{( x ^{2} + 2 ) ^{2}}

= \int \frac{1}{x + 3} + \frac{- x + 3}{x ^{2} + 2} - \frac{5 x}{( x ^{2} + 2 ) ^{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 + 3} d x + \int \frac{- x + 3}{x ^{2} + 2} d x - \int \frac{5 x}{( x ^{2} + 2 ) ^{2}} d x

\int \frac{1}{x + 3} d x = ln ∣ x + 3 ∣

\int \frac{- x + 3}{x ^{2} + 2} d x = - \frac{1}{2} ln x^{2} + 2 + \frac{3}{2} arctan (\frac{x}{2})

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

= ln ∣ x + 3 ∣ - \frac{1}{2} ln x^{2} + 2 + \frac{3}{2} arctan (\frac{x}{2}) - (- \frac{5}{2 ( x ^{2} + 2 )})

簡素化

= ln ∣ x + 3 ∣ - \frac{1}{2} ln x^{2} + 2 + \frac{3}{2} arctan (\frac{x}{2}) + \frac{5}{2 ( x ^{2} + 2 )}

定数を解答に追加する

= ln ∣ x + 3 ∣ - \frac{1}{2} ln x^{2} + 2 + \frac{3}{2} arctan (\frac{x}{2}) + \frac{5}{2 ( x ^{2} + 2 )} + C