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受欢迎的微积分 >

integral of (x^2)/(x-8)

解答

\int \frac{x ^{2}}{x - 8} d x

解答

\frac{( x - 8 ) ^{2}}{2} + 16 (x - 8) + 64 ln ∣ x - 8 ∣ + C

求解步骤

\int \frac{x ^{2}}{x - 8} d x

使用换元积分法

= \int \frac{( u + 8 ) ^{2}}{u} d u

乘开

\frac{( u + 8 ) ^{2}}{u} : u + 16 + \frac{64}{u}

= \int u + 16 + \frac{64}{u} 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 u d u + \int 16 d u + \int \frac{64}{u} d u

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

\int 16 d u = 16 u

\int \frac{64}{u} d u = 64 ln ∣ u ∣

= \frac{u ^{2}}{2} + 16 u + 64 ln ∣ u ∣

u = x - 8

代回

= \frac{( x - 8 ) ^{2}}{2} + 16 (x - 8) + 64 ln ∣ x - 8 ∣

解答补常数

= \frac{( x - 8 ) ^{2}}{2} + 16 (x - 8) + 64 ln ∣ x - 8 ∣ + C

作图

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