integral of (3x^2)/((x-3)^3)

解答

\int \frac{3 x ^{2}}{( x - 3 ) ^{3}} d x

3 (ln ∣ x - 3 ∣ - \frac{6}{x - 3} - \frac{9}{2 ( x - 3 ) ^{2}}) + C

求解步骤

\int \frac{3 x ^{2}}{( x - 3 ) ^{3}} d x

提出常数:

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

= 3 \cdot \int \frac{x ^{2}}{( x - 3 ) ^{3}} d x

使用换元积分法

= 3 \cdot \int \frac{( u + 3 ) ^{2}}{u ^{3}} d u

乘开

\frac{( u + 3 ) ^{2}}{u ^{3}} : \frac{1}{u} + \frac{6}{u ^{2}} + \frac{9}{u ^{3}}

= 3 \cdot \int \frac{1}{u} + \frac{6}{u ^{2}} + \frac{9}{u ^{3}} 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

= 3 (\int \frac{1}{u} d u + \int \frac{6}{u ^{2}} d u + \int \frac{9}{u ^{3}} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{6}{u ^{2}} d u = - \frac{6}{u}

\int \frac{9}{u ^{3}} d u = - \frac{9}{2 u ^{2}}

= 3 (ln ∣ u ∣ - \frac{6}{u} - \frac{9}{2 u ^{2}})

u = x - 3

代回

= 3 (ln ∣ x - 3 ∣ - \frac{6}{x - 3} - \frac{9}{2 ( x - 3 ) ^{2}})

解答补常数

= 3 (ln ∣ x - 3 ∣ - \frac{6}{x - 3} - \frac{9}{2 ( x - 3 ) ^{2}}) + C

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