integral of (4x-36)/(sqrt(x)+3)

解答

\int \frac{4 x - 36}{x + 3} d x

- 12 x + \frac{8}{3} x^{\frac{3}{2}} - 216 + C

求解步骤

\int \frac{4 x - 36}{x + 3} d x

乘开

\frac{4 x - 36}{x + 3} : \frac{4 x}{x + 3} - \frac{36}{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}{x + 3} d x - \int \frac{36}{x + 3} d x

\int \frac{4 x}{x + 3} d x = 8 (\frac{1}{3} x^{\frac{3}{2}} - \frac{3 x}{2} + 9 x - 27 ln x + 3)

\int \frac{36}{x + 3} d x = 72 (x + 3 - 3 ln x + 3)

= 8 (\frac{1}{3} x^{\frac{3}{2}} - \frac{3 x}{2} + 9 x - 27 ln x + 3) - 72 (x + 3 - 3 ln x + 3)

化简

8 (\frac{1}{3} x^{\frac{3}{2}} - \frac{3 x}{2} + 9 x - 27 ln x + 3) - 72 (x + 3 - 3 ln x + 3) : - 12 x + \frac{8}{3} x^{\frac{3}{2}} - 216

= - 12 x + \frac{8}{3} x^{\frac{3}{2}} - 216

解答补常数

= - 12 x + \frac{8}{3} x^{\frac{3}{2}} - 216 + C

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