integral of (1/x-2/(x^2)+5/(x^3))

解答

\int (\frac{1}{x} - \frac{2}{x ^{2}} + \frac{5}{x ^{3}}) d x

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

求解步骤

\int \frac{1}{x} - \frac{2}{x ^{2}} + \frac{5}{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{1}{x} d x - \int \frac{2}{x ^{2}} d x + \int \frac{5}{x ^{3}} d x

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

= ln ∣ x ∣ - (- \frac{2}{x}) - \frac{5}{2 x ^{2}}

化简

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

解答补常数

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

l a pl a ce t r an s f or m - y

x \to 0 lim (\frac{12}{x ^{2}})

y^{^{''}} + y^{^{'}} + a^{2} y = 0

\int \frac{1 + 6 x}{( 4 x - 3 ) ( 2 x + 5 )} d x

5 x y^{^{'}} = 2 x e^{x} - 5 y + 6 x^{2}