integral of (5x^2)/((x-1)(x^2+9))

解

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

5 (\frac{1}{10} ln ∣ x - 1 ∣ + \frac{1}{20} (9 ln x^{2} + 9 + 6 arctan (\frac{x}{3}))) + C

解答ステップ

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

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

以下の部分分数を得る:

\frac{x ^{2}}{( x - 1 ) ( x ^{2} + 9 )} : \frac{1}{10 ( x - 1 )} + \frac{9 x + 9}{10 ( x ^{2} + 9 )}

= 5 \cdot \int \frac{1}{10 ( x - 1 )} + \frac{9 x + 9}{10 ( x ^{2} + 9 )} 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

= 5 (\int \frac{1}{10 ( x - 1 )} d x + \int \frac{9 x + 9}{10 ( x ^{2} + 9 )} d x)

\int \frac{1}{10 ( x - 1 )} d x = \frac{1}{10} ln ∣ x - 1 ∣

\int \frac{9 x + 9}{10 ( x ^{2} + 9 )} d x = \frac{1}{20} (9 ln x^{2} + 9 + 6 arctan (\frac{x}{3}))

= 5 (\frac{1}{10} ln ∣ x - 1 ∣ + \frac{1}{20} (9 ln x^{2} + 9 + 6 arctan (\frac{x}{3})))

定数を解答に追加する

= 5 (\frac{1}{10} ln ∣ x - 1 ∣ + \frac{1}{20} (9 ln x^{2} + 9 + 6 arctan (\frac{x}{3}))) + C