integral of (x-1)/((2x+5)(x-8))

解答

\int \frac{x - 1}{( 2 x + 5 ) ( x - 8 )} d x

\frac{1}{6} ln ∣ 2 x + 5 ∣ + \frac{1}{3} ln ∣ x - 8 ∣ + C

求解步骤

\int \frac{x - 1}{( 2 x + 5 ) ( x - 8 )} d x

乘开

\frac{x - 1}{( 2 x + 5 ) ( x - 8 )} : \frac{x}{( 2 x + 5 ) ( x - 8 )} - \frac{1}{( 2 x + 5 ) ( 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{x}{( 2 x + 5 ) ( x - 8 )} d x - \int \frac{1}{( 2 x + 5 ) ( x - 8 )} d x

\int \frac{x}{( 2 x + 5 ) ( x - 8 )} d x = \frac{5}{42} ln ∣ 2 x + 5 ∣ + \frac{8}{21} ln ∣ x - 8 ∣

\int \frac{1}{( 2 x + 5 ) ( x - 8 )} d x = - \frac{1}{21} ln ∣ 2 x + 5 ∣ + \frac{1}{21} ln ∣ x - 8 ∣

= \frac{5}{42} ln ∣ 2 x + 5 ∣ + \frac{8}{21} ln ∣ x - 8 ∣ - (- \frac{1}{21} ln ∣ 2 x + 5 ∣ + \frac{1}{21} ln ∣ x - 8 ∣)

化简

\frac{5}{42} ln ∣ 2 x + 5 ∣ + \frac{8}{21} ln ∣ x - 8 ∣ - (- \frac{1}{21} ln ∣ 2 x + 5 ∣ + \frac{1}{21} ln ∣ x - 8 ∣) : \frac{1}{6} ln ∣ 2 x + 5 ∣ + \frac{1}{3} ln ∣ x - 8 ∣

= \frac{1}{6} ln ∣ 2 x + 5 ∣ + \frac{1}{3} ln ∣ x - 8 ∣

解答补常数

= \frac{1}{6} ln ∣ 2 x + 5 ∣ + \frac{1}{3} ln ∣ x - 8 ∣ + C

\int \frac{1}{t ^{2} 289 - t ^{2}} d t

\int_{- 7}^{7} \frac{1}{2} (49 - x^{2}) d x

t \to π lim (sin (\frac{t}{2}))

\int \frac{x}{3 x + 10} d x

in v erse l a pl a ce \frac{10}{s \cdot ( s + 1 ) \cdot ( s + 10 )}