integral of (2x^2+x+15)/(x+3)

解答

\int \frac{2 x ^{2} + x + 15}{x + 3} d x

x^{2} - 5 x + 30 ln ∣ x + 3 ∣ - 24 + C

求解步骤

\int \frac{2 x ^{2} + x + 15}{x + 3} d x

乘开

\frac{2 x ^{2} + x + 15}{x + 3} : \frac{2 x ^{2}}{x + 3} + \frac{x}{x + 3} + \frac{15}{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{2 x ^{2}}{x + 3} d x + \int \frac{x}{x + 3} d x + \int \frac{15}{x + 3} d x

\int \frac{2 x ^{2}}{x + 3} d x = x^{2} - 6 x + 18 ln ∣ x + 3 ∣ - 27

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

\int \frac{15}{x + 3} d x = 15 ln ∣ x + 3 ∣

= x^{2} - 6 x + 18 ln ∣ x + 3 ∣ - 27 + x + 3 - 3 ln ∣ x + 3 ∣ + 15 ln ∣ x + 3 ∣

化简

= x^{2} - 5 x + 30 ln ∣ x + 3 ∣ - 24

解答补常数

= x^{2} - 5 x + 30 ln ∣ x + 3 ∣ - 24 + C

\int_{1}^{2} 2 - x d x

y^{^{'}} = x y + 3

x \to - \infty lim (2 + e^{x})

\frac{d}{d x} (0.1 x)

t an g e n t y = x^{2} + 2, (2, 6)