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

解

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

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

解答ステップ

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

拡張

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

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

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

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

定数を解答に追加する

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