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

解

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

\frac{1}{81} (- \frac{1}{2} ln x^{2} + 5 - 45 arctan (\frac{x}{5})) + \frac{1}{81} ln ∣ x - 2 ∣ - \frac{2}{9 ( x - 2 )} + C

解答ステップ

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

以下の部分分数を得る:

\frac{x}{( x ^{2} + 5 ) ( x - 2 ) ^{2}} : \frac{- x - 20}{81 ( x ^{2} + 5 )} + \frac{1}{81 ( x - 2 )} + \frac{2}{9 ( x - 2 ) ^{2}}

= \int \frac{- x - 20}{81 ( x ^{2} + 5 )} + \frac{1}{81 ( x - 2 )} + \frac{2}{9 ( x - 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{- x - 20}{81 ( x ^{2} + 5 )} d x + \int \frac{1}{81 ( x - 2 )} d x + \int \frac{2}{9 ( x - 2 ) ^{2}} d x

\int \frac{- x - 20}{81 ( x ^{2} + 5 )} d x = \frac{1}{81} (- \frac{1}{2} ln x^{2} + 5 - 45 arctan (\frac{x}{5}))

\int \frac{1}{81 ( x - 2 )} d x = \frac{1}{81} ln ∣ x - 2 ∣

\int \frac{2}{9 ( x - 2 ) ^{2}} d x = - \frac{2}{9 ( x - 2 )}

= \frac{1}{81} (- \frac{1}{2} ln x^{2} + 5 - 45 arctan (\frac{x}{5})) + \frac{1}{81} ln ∣ x - 2 ∣ - \frac{2}{9 ( x - 2 )}

定数を解答に追加する

= \frac{1}{81} (- \frac{1}{2} ln x^{2} + 5 - 45 arctan (\frac{x}{5})) + \frac{1}{81} ln ∣ x - 2 ∣ - \frac{2}{9 ( x - 2 )} + C