integral of (3-sqrt(3x^2+2))/(3x^2+2)

解

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

\frac{3}{2} arctan (\frac{3}{2} x) - \frac{1}{3} ln \frac{3}{2} x + \frac{2 + 3 x ^{2}}{2} + C

解答ステップ

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

拡張

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

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

\int \frac{3 x ^{2} + 2}{3 x ^{2} + 2} d x = \frac{1}{3} ln \frac{3}{2} x + \frac{1}{2} (2 + 3 x^{2})

= \frac{3}{2} arctan (\frac{3}{2} x) - \frac{1}{3} ln \frac{3}{2} x + \frac{1}{2} (2 + 3 x^{2})

簡素化

\frac{3}{2} arctan (\frac{3}{2} x) - \frac{1}{3} ln \frac{3}{2} x + \frac{1}{2} (2 + 3 x^{2}) : \frac{3}{2} arctan (\frac{3}{2} x) - \frac{1}{3} ln \frac{3}{2} x + \frac{2 + 3 x ^{2}}{2}

= \frac{3}{2} arctan (\frac{3}{2} x) - \frac{1}{3} ln \frac{3}{2} x + \frac{2 + 3 x ^{2}}{2}

定数を解答に追加する

= \frac{3}{2} arctan (\frac{3}{2} x) - \frac{1}{3} ln \frac{3}{2} x + \frac{2 + 3 x ^{2}}{2} + C