integral of (x^2)/(x^2+x-6)

解答

\int \frac{x ^{2}}{x ^{2} + x - 6} d x

x + \frac{4}{5} ln ∣ x - 2 ∣ - \frac{9}{5} ln ∣ x + 3 ∣ + C

求解步骤

\int \frac{x ^{2}}{x ^{2} + x - 6} d x

将

\frac{x ^{2}}{x ^{2} + x - 6}

用部份分式展开:

1 + \frac{4}{5 ( x - 2 )} - \frac{9}{5 ( x + 3 )}

= \int 1 + \frac{4}{5 ( x - 2 )} - \frac{9}{5 ( x + 3 )} d x

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 1 d x + \int \frac{4}{5 ( x - 2 )} d x - \int \frac{9}{5 ( x + 3 )} d x

\int 1 d x = x

\int \frac{4}{5 ( x - 2 )} d x = \frac{4}{5} ln ∣ x - 2 ∣

\int \frac{9}{5 ( x + 3 )} d x = \frac{9}{5} ln ∣ x + 3 ∣

= x + \frac{4}{5} ln ∣ x - 2 ∣ - \frac{9}{5} ln ∣ x + 3 ∣

解答补常数

= x + \frac{4}{5} ln ∣ x - 2 ∣ - \frac{9}{5} ln ∣ x + 3 ∣ + C

x \to 0 lim (\frac{sin ( - x )}{x})

\frac{d}{d x} (e^{x} + cos (x))

\frac{d}{d x} (10 lo g_{10} (100 x))

\frac{d}{d x} (cos (2 x^{2} - 3))

d er i v a t i v e f (x) = \frac{x ^{2}}{2 + x}