積分記号 ((x-1))/((x^2+2x+3))

解

積分記号

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

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

解答ステップ

\int \frac{x - 1}{x ^{2} + 2 x + 3} d x

拡張

\frac{x - 1}{x ^{2} + 2 x + 3} : \frac{x}{x ^{2} + 2 x + 3} - \frac{1}{x ^{2} + 2 x + 3}

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}{x ^{2} + 2 x + 3} d x - \int \frac{1}{x ^{2} + 2 x + 3} d x

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

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

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

定数を解答に追加する

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