integral of (4x^2+21x+54)/(x^2+6x+13)

解

\int \frac{4 x ^{2} + 21 x + 54}{x ^{2} + 6 x + 13} d x

4 x - \frac{3}{2} ln x^{2} + 6 x + 13 + \frac{11}{2} arctan (\frac{1}{2} (x + 3)) + 12 + C

解答ステップ

\int \frac{4 x ^{2} + 21 x + 54}{x ^{2} + 6 x + 13} d x

拡張

\frac{4 x ^{2} + 21 x + 54}{x ^{2} + 6 x + 13} : \frac{4 x ^{2}}{x ^{2} + 6 x + 13} + \frac{21 x}{x ^{2} + 6 x + 13} + \frac{54}{x ^{2} + 6 x + 13}

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 ^{2} + 6 x + 13} d x + \int \frac{21 x}{x ^{2} + 6 x + 13} d x + \int \frac{54}{x ^{2} + 6 x + 13} d x

\int \frac{4 x ^{2}}{x ^{2} + 6 x + 13} d x = 4 (x + 3 - 3 ln x^{2} + 6 x + 13 + \frac{5}{2} arctan (\frac{1}{2} (x + 3)))

\int \frac{21 x}{x ^{2} + 6 x + 13} d x = 21 (\frac{1}{2} ln x^{2} + 6 x + 13 - \frac{3}{2} arctan (\frac{1}{2} (x + 3)))

\int \frac{54}{x ^{2} + 6 x + 13} d x = 27 arctan (\frac{1}{2} (x + 3))

= 4 (x + 3 - 3 ln x^{2} + 6 x + 13 + \frac{5}{2} arctan (\frac{1}{2} (x + 3))) + 21 (\frac{1}{2} ln x^{2} + 6 x + 13 - \frac{3}{2} arctan (\frac{1}{2} (x + 3))) + 27 arctan (\frac{1}{2} (x + 3))

簡素化

4 (x + 3 - 3 ln x^{2} + 6 x + 13 + \frac{5}{2} arctan (\frac{1}{2} (x + 3))) + 21 (\frac{1}{2} ln x^{2} + 6 x + 13 - \frac{3}{2} arctan (\frac{1}{2} (x + 3))) + 27 arctan (\frac{1}{2} (x + 3)) : 4 x - \frac{3}{2} ln x^{2} + 6 x + 13 + \frac{11}{2} arctan (\frac{1}{2} (x + 3)) + 12

= 4 x - \frac{3}{2} ln x^{2} + 6 x + 13 + \frac{11}{2} arctan (\frac{1}{2} (x + 3)) + 12

定数を解答に追加する

= 4 x - \frac{3}{2} ln x^{2} + 6 x + 13 + \frac{11}{2} arctan (\frac{1}{2} (x + 3)) + 12 + C