integral of (3x-2)/(5x^2-3x+8)

解

\int \frac{3 x - 2}{5 x ^{2} - 3 x + 8} d x

\frac{3}{10} ln 100 x^{2} - 60 x + 160 - \frac{11}{5 151} arctan (\frac{1}{151} (10 x - 3)) + C

解答ステップ

\int \frac{3 x - 2}{5 x ^{2} - 3 x + 8} d x

拡張

\frac{3 x - 2}{5 x ^{2} - 3 x + 8} : \frac{3 x}{5 x ^{2} - 3 x + 8} - \frac{2}{5 x ^{2} - 3 x + 8}

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{3 x}{5 x ^{2} - 3 x + 8} d x - \int \frac{2}{5 x ^{2} - 3 x + 8} d x

\int \frac{3 x}{5 x ^{2} - 3 x + 8} d x = 6 (\frac{1}{20} ln 100 x^{2} - 60 x + 160 + \frac{3}{10 151} arctan (\frac{10 x - 3}{151}))

\int \frac{2}{5 x ^{2} - 3 x + 8} d x = \frac{4}{151} arctan (\frac{10 x - 3}{151})

= 6 (\frac{1}{20} ln 100 x^{2} - 60 x + 160 + \frac{3}{10 151} arctan (\frac{10 x - 3}{151})) - \frac{4}{151} arctan (\frac{10 x - 3}{151})

簡素化

6 (\frac{1}{20} ln 100 x^{2} - 60 x + 160 + \frac{3}{10 151} arctan (\frac{10 x - 3}{151})) - \frac{4}{151} arctan (\frac{10 x - 3}{151}) : \frac{3}{10} ln 100 x^{2} - 60 x + 160 - \frac{11}{5 151} arctan (\frac{1}{151} (10 x - 3))

= \frac{3}{10} ln 100 x^{2} - 60 x + 160 - \frac{11}{5 151} arctan (\frac{1}{151} (10 x - 3))

定数を解答に追加する

= \frac{3}{10} ln 100 x^{2} - 60 x + 160 - \frac{11}{5 151} arctan (\frac{1}{151} (10 x - 3)) + C