integral of (2x^2-24)/(x(x+2)^3)

解答

\int \frac{2 x ^{2} - 24}{x ( x + 2 ) ^{3}} d x

- \frac{8}{x + 2} - \frac{4}{( x + 2 ) ^{2}} - 3 ln (\frac{1}{2} ∣ x ∣) + 3 ln \frac{x}{2} + 1 + C

求解步骤

\int \frac{2 x ^{2} - 24}{x ( x + 2 ) ^{3}} d x

乘开

\frac{2 x ^{2} - 24}{x ( x + 2 ) ^{3}} : \frac{2 x}{( x + 2 ) ^{3}} - \frac{24}{x ( x + 2 ) ^{3}}

= \int \frac{2 x}{( x + 2 ) ^{3}} - \frac{24}{x ( x + 2 ) ^{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{2 x}{( x + 2 ) ^{3}} d x - \int \frac{24}{x ( x + 2 ) ^{3}} d x

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

\int \frac{24}{x ( x + 2 ) ^{3}} d x = 6 (\frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} + \frac{1}{2} ln (\frac{1}{2} ∣ x ∣) - \frac{1}{2} ln \frac{x}{2} + 1)

= 2 (- \frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}}) - 6 (\frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} + \frac{1}{2} ln (\frac{1}{2} ∣ x ∣) - \frac{1}{2} ln \frac{x}{2} + 1)

化简

2 (- \frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}}) - 6 (\frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} + \frac{1}{2} ln (\frac{1}{2} ∣ x ∣) - \frac{1}{2} ln \frac{x}{2} + 1) : - \frac{8}{x + 2} - \frac{4}{( x + 2 ) ^{2}} - 3 ln (\frac{1}{2} ∣ x ∣) + 3 ln \frac{x}{2} + 1

= - \frac{8}{x + 2} - \frac{4}{( x + 2 ) ^{2}} - 3 ln (\frac{1}{2} ∣ x ∣) + 3 ln \frac{x}{2} + 1

解答补常数

= - \frac{8}{x + 2} - \frac{4}{( x + 2 ) ^{2}} - 3 ln (\frac{1}{2} ∣ x ∣) + 3 ln \frac{x}{2} + 1 + C