integral of (4x^2+x-2)/(x-3)

解答

\int \frac{4 x ^{2} + x - 2}{x - 3} d x

2 x^{2} + 13 x + 37 ln ∣ x - 3 ∣ - 57 + C

求解步骤

\int \frac{4 x ^{2} + x - 2}{x - 3} d x

乘开

\frac{4 x ^{2} + x - 2}{x - 3} : \frac{4 x ^{2}}{x - 3} + \frac{x}{x - 3} - \frac{2}{x - 3}

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 \frac{4 x ^{2}}{x - 3} d x + \int \frac{x}{x - 3} d x - \int \frac{2}{x - 3} d x

\int \frac{4 x ^{2}}{x - 3} d x = 4 (\frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣)

\int \frac{x}{x - 3} d x = x - 3 + 3 ln ∣ x - 3 ∣

\int \frac{2}{x - 3} d x = 2 ln ∣ x - 3 ∣

= 4 (\frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣) + x - 3 + 3 ln ∣ x - 3 ∣ - 2 ln ∣ x - 3 ∣

化简

4 (\frac{( x - 3 ) ^{2}}{2} + 6 (x - 3) + 9 ln ∣ x - 3 ∣) + x - 3 + 3 ln ∣ x - 3 ∣ - 2 ln ∣ x - 3 ∣ : 2 x^{2} + 13 x + 37 ln ∣ x - 3 ∣ - 57

= 2 x^{2} + 13 x + 37 ln ∣ x - 3 ∣ - 57

解答补常数

= 2 x^{2} + 13 x + 37 ln ∣ x - 3 ∣ - 57 + C