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受欢迎的微积分 >

integral of (x^2+2x-1)/(3-x)

解答

\int \frac{x ^{2} + 2 x - 1}{3 - x} d x

解答

- \frac{1}{2} (3 - x)^{2} + 24 - 8 x - 14 ln ∣ 3 - x ∣ + C

求解步骤

\int \frac{x ^{2} + 2 x - 1}{3 - x} d x

使用换元积分法

= \int - \frac{u ^{2} - 8 u + 14}{u} d u

提出常数:

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

= - \int \frac{u ^{2} - 8 u + 14}{u} d u

乘开

\frac{u ^{2} - 8 u + 14}{u} : u - 8 + \frac{14}{u}

= - \int u - 8 + \frac{14}{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 8 d u + \int \frac{14}{u} d u)

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

\int 8 d u = 8 u

\int \frac{14}{u} d u = 14 ln ∣ u ∣

= - (\frac{u ^{2}}{2} - 8 u + 14 ln ∣ u ∣)

u = 3 - x

代回

= - (\frac{( 3 - x ) ^{2}}{2} - 8 (3 - x) + 14 ln ∣ 3 - x ∣)

化简

- (\frac{( 3 - x ) ^{2}}{2} - 8 (3 - x) + 14 ln ∣ 3 - x ∣) : - \frac{1}{2} (3 - x)^{2} + 24 - 8 x - 14 ln ∣ 3 - x ∣

= - \frac{1}{2} (3 - x)^{2} + 24 - 8 x - 14 ln ∣ 3 - x ∣

解答补常数

= - \frac{1}{2} (3 - x)^{2} + 24 - 8 x - 14 ln ∣ 3 - x ∣ + C

作图

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