integral of (x^3-4x+3)/(x+5)

解答

\int \frac{x ^{3} - 4 x + 3}{x + 5} d x

x^{3} + 15 x^{2} + 146 x + \frac{1}{3} (x + 5)^{3} - 102 ln ∣ x + 5 ∣ - (x + 5)^{3} - \frac{15}{2} (x + 5)^{2} + 378 + C

求解步骤

\int \frac{x ^{3} - 4 x + 3}{x + 5} d x

使用换元积分法

= \int \frac{- 4 u + ( u - 5 ) ^{3} + 23}{u} d u

使用分布积分法

= - 4 u + (u - 5)^{3} + 23 - \int \frac{3 u ^{3} - 30 u ^{2} + 75 u - ( u - 5 ) ^{3} - 23}{u} d u

\int \frac{3 u ^{3} - 30 u ^{2} + 75 u - ( u - 5 ) ^{3} - 23}{u} d u = u^{3} - \frac{u ^{3}}{3} - 15 u^{2} + \frac{15 u ^{2}}{2} + 102 ln ∣ u ∣

= - 4 u + (u - 5)^{3} + 23 - (u^{3} - \frac{u ^{3}}{3} - 15 u^{2} + \frac{15 u ^{2}}{2} + 102 ln ∣ u ∣)

u = x + 5

代回

= - 4 (x + 5) + (x + 5 - 5)^{3} + 23 - ((x + 5)^{3} - \frac{( x + 5 ) ^{3}}{3} - 15 (x + 5)^{2} + \frac{15 ( x + 5 ) ^{2}}{2} + 102 ln ∣ x + 5 ∣)

化简

- 4 (x + 5) + (x + 5 - 5)^{3} + 23 - ((x + 5)^{3} - \frac{( x + 5 ) ^{3}}{3} - 15 (x + 5)^{2} + \frac{15 ( x + 5 ) ^{2}}{2} + 102 ln ∣ x + 5 ∣) : x^{3} + 15 x^{2} + 146 x + \frac{1}{3} (x + 5)^{3} - 102 ln ∣ x + 5 ∣ - (x + 5)^{3} - \frac{15}{2} (x + 5)^{2} + 378

= x^{3} + 15 x^{2} + 146 x + \frac{1}{3} (x + 5)^{3} - 102 ln ∣ x + 5 ∣ - (x + 5)^{3} - \frac{15}{2} (x + 5)^{2} + 378

解答补常数

= x^{3} + 15 x^{2} + 146 x + \frac{1}{3} (x + 5)^{3} - 102 ln ∣ x + 5 ∣ - (x + 5)^{3} - \frac{15}{2} (x + 5)^{2} + 378 + C

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