integral of (4x^2+16x+16)/(x^3+8x^2+16x)

解

\int \frac{4 x ^{2} + 16 x + 16}{x ^{3} + 8 x ^{2} + 16 x} d x

3 ln ∣ x + 4 ∣ + \frac{4}{x + 4} + ln ∣ x ∣ + C

解答ステップ

\int \frac{4 x ^{2} + 16 x + 16}{x ^{3} + 8 x ^{2} + 16 x} d x

拡張

\frac{4 x ^{2} + 16 x + 16}{x ^{3} + 8 x ^{2} + 16 x} : \frac{4 x ^{2}}{x ^{3} + 8 x ^{2} + 16 x} + \frac{16 x}{x ^{3} + 8 x ^{2} + 16 x} + \frac{16}{x ^{3} + 8 x ^{2} + 16 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{4 x ^{2}}{x ^{3} + 8 x ^{2} + 16 x} d x + \int \frac{16 x}{x ^{3} + 8 x ^{2} + 16 x} d x + \int \frac{16}{x ^{3} + 8 x ^{2} + 16 x} d x

\int \frac{4 x ^{2}}{x ^{3} + 8 x ^{2} + 16 x} d x = 4 (ln ∣ x + 4 ∣ + \frac{4}{x + 4})

\int \frac{16 x}{x ^{3} + 8 x ^{2} + 16 x} d x = - \frac{16}{x + 4}

\int \frac{16}{x ^{3} + 8 x ^{2} + 16 x} d x = ln ∣ x ∣ - ln ∣ x + 4 ∣ + \frac{4}{x + 4}

= 4 (ln ∣ x + 4 ∣ + \frac{4}{x + 4}) - \frac{16}{x + 4} + ln ∣ x ∣ - ln ∣ x + 4 ∣ + \frac{4}{x + 4}

簡素化

4 (ln ∣ x + 4 ∣ + \frac{4}{x + 4}) - \frac{16}{x + 4} + ln ∣ x ∣ - ln ∣ x + 4 ∣ + \frac{4}{x + 4} : 3 ln ∣ x + 4 ∣ + \frac{4}{x + 4} + ln ∣ x ∣

= 3 ln ∣ x + 4 ∣ + \frac{4}{x + 4} + ln ∣ x ∣

定数を解答に追加する

= 3 ln ∣ x + 4 ∣ + \frac{4}{x + 4} + ln ∣ x ∣ + C