integral of (x^3+2x^2+5x+3)/(x^2+x+2)

解

\int \frac{x ^{3} + 2 x ^{2} + 5 x + 3}{x ^{2} + x + 2} d x

\frac{x ^{2}}{2} + x + ln 4 x^{2} + 4 x + 8 + \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) - \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) + C

解答ステップ

\int \frac{x ^{3} + 2 x ^{2} + 5 x + 3}{x ^{2} + x + 2} d x

拡張

\frac{x ^{3} + 2 x ^{2} + 5 x + 3}{x ^{2} + x + 2} : \frac{x ^{3}}{x ^{2} + x + 2} + \frac{2 x ^{2}}{x ^{2} + x + 2} + \frac{5 x}{x ^{2} + x + 2} + \frac{3}{x ^{2} + x + 2}

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 ^{2} + x + 2} d x + \int \frac{2 x ^{2}}{x ^{2} + x + 2} d x + \int \frac{5 x}{x ^{2} + x + 2} d x + \int \frac{3}{x ^{2} + x + 2} d x

\int \frac{x ^{3}}{x ^{2} + x + 2} d x = \frac{x ^{2}}{2} - x - \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{5}{7} arctan (\frac{2 x + 1}{7})

\int \frac{2 x ^{2}}{x ^{2} + x + 2} d x = - ln 4 x^{2} + 4 x + 8 - \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) + 2 x

\int \frac{5 x}{x ^{2} + x + 2} d x = 10 (\frac{1}{4} ln 4 x^{2} + 4 x + 8 - \frac{1}{2 7} arctan (\frac{2 x + 1}{7}))

\int \frac{3}{x ^{2} + x + 2} d x = \frac{6}{7} arctan (\frac{2 x + 1}{7})

= \frac{x ^{2}}{2} - x - \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{5}{7} arctan (\frac{2 x + 1}{7}) - ln 4 x^{2} + 4 x + 8 - \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) + 2 x + 10 (\frac{1}{4} ln 4 x^{2} + 4 x + 8 - \frac{1}{2 7} arctan (\frac{2 x + 1}{7})) + \frac{6}{7} arctan (\frac{2 x + 1}{7})

簡素化

\frac{x ^{2}}{2} - x - \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{5}{7} arctan (\frac{2 x + 1}{7}) - ln 4 x^{2} + 4 x + 8 - \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) + 2 x + 10 (\frac{1}{4} ln 4 x^{2} + 4 x + 8 - \frac{1}{2 7} arctan (\frac{2 x + 1}{7})) + \frac{6}{7} arctan (\frac{2 x + 1}{7}) : \frac{x ^{2}}{2} + x + ln 4 x^{2} + 4 x + 8 + \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) - \frac{6}{7} arctan (\frac{1}{7} (2 x + 1))

= \frac{x ^{2}}{2} + x + ln 4 x^{2} + 4 x + 8 + \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) - \frac{6}{7} arctan (\frac{1}{7} (2 x + 1))

定数を解答に追加する

= \frac{x ^{2}}{2} + x + ln 4 x^{2} + 4 x + 8 + \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) - \frac{6}{7} arctan (\frac{1}{7} (2 x + 1)) + C