integral of (4+x^2)/(x^3)

解答

\int \frac{4 + x ^{2}}{x ^{3}} d x

- \frac{2}{x ^{2}} + ln ∣ x ∣ + C

求解步骤

\int \frac{4 + x ^{2}}{x ^{3}} d x

乘开

\frac{4 + x ^{2}}{x ^{3}} : \frac{4}{x ^{3}} + \frac{1}{x}

= \int \frac{4}{x ^{3}} + \frac{1}{x} 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{4}{x ^{3}} d x + \int \frac{1}{x} d x

\int \frac{4}{x ^{3}} d x = - \frac{2}{x ^{2}}

\int \frac{1}{x} d x = ln ∣ x ∣

= - \frac{2}{x ^{2}} + ln ∣ x ∣

解答补常数

= - \frac{2}{x ^{2}} + ln ∣ x ∣ + C

\frac{d Q}{d t} = 3 - \frac{3 Q}{1000}

x \to - 1 lim (∣ x ∣ - x)

(2 x + 5) y^{^{'}} + 10 y = 10 (2 x + 5)

n = 0 \sum \infty \frac{3}{5 ^{n}}

\int (x^{2} - 2 x) d x