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

解答

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

3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) + C

求解步骤

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

使用换元积分法

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

乘开

\frac{( u + 2 ) ^{2}}{u} : u + 4 + \frac{4}{u}

= 3 \cdot \int u + 4 + \frac{4}{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

= 3 (\int u d u + \int 4 d u + \int \frac{4}{u} d u)

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

\int 4 d u = 4 u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

= 3 (\frac{u ^{2}}{2} + 4 u + 4 ln ∣ u ∣)

u = x - 2

代回

= 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣)

解答补常数

= 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) + C

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