integral of (144)/(x^3-12x^2)

解答

\int \frac{144}{x ^{3} - 12 x ^{2}} d x

- ln ∣ x ∣ + \frac{12}{x} + ln ∣ x - 12 ∣ + C

求解步骤

\int \frac{144}{x ^{3} - 12 x ^{2}} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 144 \cdot \int \frac{1}{x ^{3} - 12 x ^{2}} d x

将

\frac{1}{x ^{3} - 12 x ^{2}}

用部份分式展开:

- \frac{1}{144 x} - \frac{1}{12 x ^{2}} + \frac{1}{144 ( x - 12 )}

= 144 \cdot \int - \frac{1}{144 x} - \frac{1}{12 x ^{2}} + \frac{1}{144 ( x - 12 )} 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

= 144 (- \int \frac{1}{144 x} d x - \int \frac{1}{12 x ^{2}} d x + \int \frac{1}{144 ( x - 12 )} d x)

\int \frac{1}{144 x} d x = \frac{1}{144} ln ∣ x ∣

\int \frac{1}{12 x ^{2}} d x = - \frac{1}{12 x}

\int \frac{1}{144 ( x - 12 )} d x = \frac{1}{144} ln ∣ x - 12 ∣

= 144 (- \frac{1}{144} ln ∣ x ∣ - (- \frac{1}{12 x}) + \frac{1}{144} ln ∣ x - 12 ∣)

化简

144 (- \frac{1}{144} ln ∣ x ∣ - (- \frac{1}{12 x}) + \frac{1}{144} ln ∣ x - 12 ∣) : - ln ∣ x ∣ + \frac{12}{x} + ln ∣ x - 12 ∣

= - ln ∣ x ∣ + \frac{12}{x} + ln ∣ x - 12 ∣

解答补常数

= - ln ∣ x ∣ + \frac{12}{x} + ln ∣ x - 12 ∣ + C

d er i v a t i v e f (2) = \frac{x + 11}{x - 9}

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

t \to 1 lim (\frac{t ^{2} - 4}{t - 2})

\int (9 x^{5} - 6 x^{3} + 4 x) d x

x \to 1 lim (\frac{1 - \frac{1}{x}}{x - 1})