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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

对

x^{2} - 2 x + 1

配方:

(x - 1)^{2}

= \int \frac{x ^{3}}{( x - 1 ) ^{2}} d x

使用换元积分法

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

乘开

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

= \int u + 3 + \frac{3}{u} + \frac{1}{u ^{2}} 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 3 d u + \int \frac{3}{u} d u + \int \frac{1}{u ^{2}} d u

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

\int 3 d u = 3 u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

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

= \frac{u ^{2}}{2} + 3 u + 3 ln ∣ u ∣ - \frac{1}{u}

u = x - 1

代回

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

解答补常数

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

作图

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