integral of (x+8)/(3x^2-12)

解答

\int \frac{x + 8}{3 x ^{2} - 12} d x

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

求解步骤

\int \frac{x + 8}{3 x ^{2} - 12} d x

将

\frac{x + 8}{3 x ^{2} - 12}

用部份分式展开:

- \frac{1}{2 ( x + 2 )} + \frac{5}{6 ( x - 2 )}

= \int - \frac{1}{2 ( x + 2 )} + \frac{5}{6 ( x - 2 )} 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{1}{2 ( x + 2 )} d x + \int \frac{5}{6 ( x - 2 )} d x

\int \frac{1}{2 ( x + 2 )} d x = \frac{1}{2} ln ∣ x + 2 ∣

\int \frac{5}{6 ( x - 2 )} d x = \frac{5}{6} ln ∣ x - 2 ∣

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

解答补常数

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

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

\int \frac{lo g _{3} ( x )}{3 x} d x

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

d er i v a t i v e tan (3) x^{2}

d er i v a t i v e 3 t - 4 t^{2}