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

解答

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

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

求解步骤

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

将

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

用部份分式展开:

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

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

\int \frac{5}{x} d x = 5 ln ∣ x ∣

\int \frac{5}{x + 1} d x = 5 ln ∣ x + 1 ∣

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

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

解答补常数

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

\frac{\partial}{\partial x} (x ln (z^{5} + x^{4}))

\frac{\partial}{\partial x} (7 x y^{2} + 1)

x \to 0 lim ((1 + 5 x)^{\frac{3}{x}})

d er i v a t i v e \frac{( x ^{4} + 2 )}{ln ( x )}

\frac{\partial}{\partial x} (e^{- x y})