integral of (x-3)/(x^2+5x+6)

解答

\int \frac{x - 3}{x ^{2} + 5 x + 6} d x

- 5 ln ∣ x + 2 ∣ + 6 ln ∣ x + 3 ∣ + C

求解步骤

\int \frac{x - 3}{x ^{2} + 5 x + 6} d x

将

\frac{x - 3}{x ^{2} + 5 x + 6}

用部份分式展开:

- \frac{5}{x + 2} + \frac{6}{x + 3}

= \int - \frac{5}{x + 2} + \frac{6}{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 \frac{5}{x + 2} d x + \int \frac{6}{x + 3} d x

\int \frac{5}{x + 2} d x = 5 ln ∣ x + 2 ∣

\int \frac{6}{x + 3} d x = 6 ln ∣ x + 3 ∣

= - 5 ln ∣ x + 2 ∣ + 6 ln ∣ x + 3 ∣

解答补常数

= - 5 ln ∣ x + 2 ∣ + 6 ln ∣ x + 3 ∣ + C

\int \frac{e ^{y}}{e ^{y} - 1} d y

\int \frac{1}{x ( 7 + ln ( x ^{2} ) )} d x

\int \frac{ln ( x e ^{x} )}{x} d x

\int \frac{3}{sec ^{2} ( 5 x )} d x

\int \frac{cos ( x )}{3 sin ^{2} ( x ) + 4 sin ( x )} d x