integral of (3x^2+2x-10)/(x^3+x^2-10x+8)

解答

\int \frac{3 x ^{2} + 2 x - 10}{x ^{3} + x ^{2} - 10 x + 8} d x

ln ∣ x - 1 ∣ + ln ∣ x + 4 ∣ + ln ∣ x - 2 ∣ + C

求解步骤

\int \frac{3 x ^{2} + 2 x - 10}{x ^{3} + x ^{2} - 10 x + 8} d x

乘开

\frac{3 x ^{2} + 2 x - 10}{x ^{3} + x ^{2} - 10 x + 8} : \frac{3 x ^{2}}{x ^{3} + x ^{2} - 10 x + 8} + \frac{2 x}{x ^{3} + x ^{2} - 10 x + 8} - \frac{10}{x ^{3} + x ^{2} - 10 x + 8}

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{3 x ^{2}}{x ^{3} + x ^{2} - 10 x + 8} d x + \int \frac{2 x}{x ^{3} + x ^{2} - 10 x + 8} d x - \int \frac{10}{x ^{3} + x ^{2} - 10 x + 8} d x

\int \frac{3 x ^{2}}{x ^{3} + x ^{2} - 10 x + 8} d x = - \frac{3}{5} ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣ + \frac{8}{5} ln ∣ x + 4 ∣

\int \frac{2 x}{x ^{3} + x ^{2} - 10 x + 8} d x = 2 (- \frac{1}{5} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x - 2 ∣ - \frac{2}{15} ln ∣ x + 4 ∣)

\int \frac{10}{x ^{3} + x ^{2} - 10 x + 8} d x = 10 (- \frac{1}{5} ln ∣ x - 1 ∣ + \frac{1}{6} ln ∣ x - 2 ∣ + \frac{1}{30} ln ∣ x + 4 ∣)

= - \frac{3}{5} ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣ + \frac{8}{5} ln ∣ x + 4 ∣ + 2 (- \frac{1}{5} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x - 2 ∣ - \frac{2}{15} ln ∣ x + 4 ∣) - 10 (- \frac{1}{5} ln ∣ x - 1 ∣ + \frac{1}{6} ln ∣ x - 2 ∣ + \frac{1}{30} ln ∣ x + 4 ∣)

化简

- \frac{3}{5} ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣ + \frac{8}{5} ln ∣ x + 4 ∣ + 2 (- \frac{1}{5} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x - 2 ∣ - \frac{2}{15} ln ∣ x + 4 ∣) - 10 (- \frac{1}{5} ln ∣ x - 1 ∣ + \frac{1}{6} ln ∣ x - 2 ∣ + \frac{1}{30} ln ∣ x + 4 ∣) : ln ∣ x - 1 ∣ + ln ∣ x + 4 ∣ + ln ∣ x - 2 ∣

= ln ∣ x - 1 ∣ + ln ∣ x + 4 ∣ + ln ∣ x - 2 ∣

解答补常数

= ln ∣ x - 1 ∣ + ln ∣ x + 4 ∣ + ln ∣ x - 2 ∣ + C

\int \frac{1}{x ln ( x )} d x

\int_{0}^{1} \frac{2 x + 3}{( x + 1 ) ^{2}} d x

\int (\frac{e ^{- \frac{2 y}{x}} y ^{3}}{x ^{4}}) d x

d er i v a t i v e \frac{3}{5} cos (2 x - \frac{π}{4}) + π

a re a x^{2} e^{- x}, x e^{- x}