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

解答

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

ln ∣ x + y ∣ + \frac{x}{x + y} + C

求解步骤

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

使用换元积分法

= \int \frac{u - x}{u ^{2}} d u

乘开

\frac{u - x}{u ^{2}} : \frac{1}{u} - \frac{x}{u ^{2}}

= \int \frac{1}{u} - \frac{x}{u ^{2}} d u

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{1}{u} d u - \int \frac{x}{u ^{2}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{x}{u ^{2}} d u = - \frac{x}{u}

= ln ∣ u ∣ - (- \frac{x}{u})

u = x + y

代回

= ln ∣ x + y ∣ - (- \frac{x}{x + y})

化简

= ln ∣ x + y ∣ + \frac{x}{x + y}

解答补常数

= ln ∣ x + y ∣ + \frac{x}{x + y} + C

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