integral of (x^3+93x+70)/(x^3+100x)

解

\int \frac{x ^{3} + 93 x + 70}{x ^{3} + 100 x} d x

x + \frac{7}{10} ln ∣ x ∣ - \frac{7}{20} ln x^{2} + 100 - \frac{7}{10} arctan (\frac{x}{10}) + C

解答ステップ

\int \frac{x ^{3} + 93 x + 70}{x ^{3} + 100 x} d x

拡張

\frac{x ^{3} + 93 x + 70}{x ^{3} + 100 x} : \frac{x ^{3}}{x ^{3} + 100 x} + \frac{93 x}{x ^{3} + 100 x} + \frac{70}{x ^{3} + 100 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 ^{3}}{x ^{3} + 100 x} d x + \int \frac{93 x}{x ^{3} + 100 x} d x + \int \frac{70}{x ^{3} + 100 x} d x

\int \frac{x ^{3}}{x ^{3} + 100 x} d x = - 10 arctan (\frac{x}{10}) + x

\int \frac{93 x}{x ^{3} + 100 x} d x = \frac{93}{10} arctan (\frac{x}{10})

\int \frac{70}{x ^{3} + 100 x} d x = 70 (\frac{1}{100} ln ∣ x ∣ - \frac{1}{200} ln x^{2} + 100)

= - 10 arctan (\frac{x}{10}) + x + \frac{93}{10} arctan (\frac{x}{10}) + 70 (\frac{1}{100} ln ∣ x ∣ - \frac{1}{200} ln x^{2} + 100)

簡素化

- 10 arctan (\frac{x}{10}) + x + \frac{93}{10} arctan (\frac{x}{10}) + 70 (\frac{1}{100} ln ∣ x ∣ - \frac{1}{200} ln x^{2} + 100) : x + \frac{7}{10} ln ∣ x ∣ - \frac{7}{20} ln x^{2} + 100 - \frac{7}{10} arctan (\frac{x}{10})

= x + \frac{7}{10} ln ∣ x ∣ - \frac{7}{20} ln x^{2} + 100 - \frac{7}{10} arctan (\frac{x}{10})

定数を解答に追加する

= x + \frac{7}{10} ln ∣ x ∣ - \frac{7}{20} ln x^{2} + 100 - \frac{7}{10} arctan (\frac{x}{10}) + C