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

解

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

\frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 2 ∣ + \frac{3}{x + 2} + C

解答ステップ

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

以下の部分分数を得る:

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

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

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

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

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

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

簡素化

= \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 2 ∣ + \frac{3}{x + 2}

定数を解答に追加する

= \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 2 ∣ + \frac{3}{x + 2} + C