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

解答

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

- \frac{5}{x + 1} + \frac{3}{2 ( x + 1 ) ^{2}} - 2 ln ∣ x ∣ + 2 ln ∣ x + 1 ∣ - \frac{1}{( x + 1 ) ^{2}} + C

求解步骤

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

乘开

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

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

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

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

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

化简

3 (- \frac{1}{x + 1} + \frac{1}{2 ( x + 1 ) ^{2}}) - (2 ln ∣ x ∣ - 2 ln ∣ x + 1 ∣ + \frac{2}{x + 1} + \frac{1}{( x + 1 ) ^{2}}) : - \frac{5}{x + 1} + \frac{3}{2 ( x + 1 ) ^{2}} - 2 ln ∣ x ∣ + 2 ln ∣ x + 1 ∣ - \frac{1}{( x + 1 ) ^{2}}

= - \frac{5}{x + 1} + \frac{3}{2 ( x + 1 ) ^{2}} - 2 ln ∣ x ∣ + 2 ln ∣ x + 1 ∣ - \frac{1}{( x + 1 ) ^{2}}

解答补常数

= - \frac{5}{x + 1} + \frac{3}{2 ( x + 1 ) ^{2}} - 2 ln ∣ x ∣ + 2 ln ∣ x + 1 ∣ - \frac{1}{( x + 1 ) ^{2}} + C

s im pl i f y lo g_{10} (\frac{( x + 1 ) ( x - 8 )}{( x - 1 ) ( x + 8 )})

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

\frac{\partial}{\partial x} (e^{8 x y})

\frac{d}{d x} (x^{k})

x \to \infty lim (- \frac{1}{3})