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

解答

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

- 2 ln ∣ x + 1 ∣ + ln ∣ x + 3 ∣ + ln ∣ x ∣ + C

求解步骤

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

乘开

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

\int \frac{3}{x ^{3} + 4 x ^{2} + 3 x} d x = ln ∣ x ∣ - \frac{3}{2} ln ∣ x + 1 ∣ + \frac{1}{2} ln ∣ x + 3 ∣

\int \frac{x}{x ^{3} + 4 x ^{2} + 3 x} d x = \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x + 3 ∣

= ln ∣ x ∣ - \frac{3}{2} ln ∣ x + 1 ∣ + \frac{1}{2} ln ∣ x + 3 ∣ - (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x + 3 ∣)

化简

ln ∣ x ∣ - \frac{3}{2} ln ∣ x + 1 ∣ + \frac{1}{2} ln ∣ x + 3 ∣ - (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x + 3 ∣) : - 2 ln ∣ x + 1 ∣ + ln ∣ x + 3 ∣ + ln ∣ x ∣

= - 2 ln ∣ x + 1 ∣ + ln ∣ x + 3 ∣ + ln ∣ x ∣

解答补常数

= - 2 ln ∣ x + 1 ∣ + ln ∣ x + 3 ∣ + ln ∣ x ∣ + C

\frac{d}{d x} (4 ln (cos (x)))

x \to 3 - lim (2 x - 1)

\int_{1}^{4} 5 t ln (t) d t

x \to - 3 lim (- \frac{5}{( x + 3 ) ^{2}})

4 y^{^{'}} = \frac{4 x y + 8 x ^{2} + 2 y ^{2}}{x ^{2}}