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

解答

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

ln ∣ x - 2 ∣ - \frac{4}{x - 2} + C

求解步骤

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

乘开

\frac{x + 2}{( x - 2 ) ^{2}} : \frac{x}{( x - 2 ) ^{2}} + \frac{2}{( x - 2 ) ^{2}}

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

\int \frac{x}{( x - 2 ) ^{2}} d x = ln ∣ x - 2 ∣ - \frac{2}{x - 2}

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

= ln ∣ x - 2 ∣ - \frac{2}{x - 2} - \frac{2}{x - 2}

化简

ln ∣ x - 2 ∣ - \frac{2}{x - 2} - \frac{2}{x - 2} : ln ∣ x - 2 ∣ - \frac{4}{x - 2}

= ln ∣ x - 2 ∣ - \frac{4}{x - 2}

解答补常数

= ln ∣ x - 2 ∣ - \frac{4}{x - 2} + C

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