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

解答

\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

求解步骤

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

乘开

\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 )}

使用积分加法定则:

\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))

解答补常数

= - \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

x \to 0 lim (x + ∣ x ∣)

\frac{d}{d x} (e^{4 x} + e^{- x})

\frac{\partial}{\partial x} (\frac{a x + b y}{c x + d y})

\frac{d}{d x} (x + \frac{1}{x})

x \to - 1 lim (- x^{2} + 4 x - 8)