integral of x/((x+a)(x-a))

解答

\int \frac{x}{( x + a ) ( x - a )} d x

\frac{1}{2} ln (∣ x + a ∣ ∣ x - a ∣) + C

求解步骤

\int \frac{x}{( x + a ) ( x - a )} d x

使用换元积分法

= \int \frac{1}{2 u} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

u = (x + a) (x - a)

代回

= \frac{1}{2} ln ∣ (x + a) (x - a) ∣

化简

\frac{1}{2} ln ∣ (x + a) (x - a) ∣ : \frac{1}{2} ln (∣ x + a ∣ ∣ x - a ∣)

= \frac{1}{2} ln (∣ x + a ∣ ∣ x - a ∣)

解答补常数

= \frac{1}{2} ln (∣ x + a ∣ ∣ x - a ∣) + C

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