integral of (x+6)/((x^2+6)(x^2+1))

解

\int \frac{x + 6}{( x ^{2} + 6 ) ( x ^{2} + 1 )} d x

- \frac{1}{10} ln x^{2} + 6 + \frac{1}{10} ln x^{2} + 1 + 6 (- \frac{1}{5 6} arctan (\frac{x}{6}) + \frac{1}{5} arctan (x)) + C

解答ステップ

\int \frac{x + 6}{( x ^{2} + 6 ) ( x ^{2} + 1 )} d x

拡張

\frac{x + 6}{( x ^{2} + 6 ) ( x ^{2} + 1 )} : \frac{x}{( x ^{2} + 6 ) ( x ^{2} + 1 )} + \frac{6}{( x ^{2} + 6 ) ( x ^{2} + 1 )}

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} + 6 ) ( x ^{2} + 1 )} d x + \int \frac{6}{( x ^{2} + 6 ) ( x ^{2} + 1 )} d x

\int \frac{x}{( x ^{2} + 6 ) ( x ^{2} + 1 )} d x = - \frac{1}{10} ln x^{2} + 6 + \frac{1}{10} ln x^{2} + 1

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

= - \frac{1}{10} ln x^{2} + 6 + \frac{1}{10} ln x^{2} + 1 + 6 (- \frac{1}{5 6} arctan (\frac{x}{6}) + \frac{1}{5} arctan (x))

定数を解答に追加する

= - \frac{1}{10} ln x^{2} + 6 + \frac{1}{10} ln x^{2} + 1 + 6 (- \frac{1}{5 6} arctan (\frac{x}{6}) + \frac{1}{5} arctan (x)) + C