integral of (8x^2-7x+16)/(x^3+4x)

解

\int \frac{8 x ^{2} - 7 x + 16}{x ^{3} + 4 x} d x

- \frac{7}{2} arctan (\frac{x}{2}) + 2 ln x^{2} + 4 + 4 ln ∣ x ∣ + C

解答ステップ

\int \frac{8 x ^{2} - 7 x + 16}{x ^{3} + 4 x} d x

拡張

\frac{8 x ^{2} - 7 x + 16}{x ^{3} + 4 x} : \frac{8 x ^{2}}{x ^{3} + 4 x} - \frac{7 x}{x ^{3} + 4 x} + \frac{16}{x ^{3} + 4 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{8 x ^{2}}{x ^{3} + 4 x} d x - \int \frac{7 x}{x ^{3} + 4 x} d x + \int \frac{16}{x ^{3} + 4 x} d x

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

\int \frac{7 x}{x ^{3} + 4 x} d x = \frac{7}{2} arctan (\frac{x}{2})

\int \frac{16}{x ^{3} + 4 x} d x = 16 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4)

= 4 ln x^{2} + 4 - \frac{7}{2} arctan (\frac{x}{2}) + 16 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4)

簡素化

4 ln x^{2} + 4 - \frac{7}{2} arctan (\frac{x}{2}) + 16 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{8} ln x^{2} + 4) : - \frac{7}{2} arctan (\frac{x}{2}) + 2 ln x^{2} + 4 + 4 ln ∣ x ∣

= - \frac{7}{2} arctan (\frac{x}{2}) + 2 ln x^{2} + 4 + 4 ln ∣ x ∣

定数を解答に追加する

= - \frac{7}{2} arctan (\frac{x}{2}) + 2 ln x^{2} + 4 + 4 ln ∣ x ∣ + C