integral of (2x^2-10x+14)/((x-3)^3)

解答

\int \frac{2 x ^{2} - 10 x + 14}{( x - 3 ) ^{3}} d x

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

求解步骤

\int \frac{2 x ^{2} - 10 x + 14}{( x - 3 ) ^{3}} d x

乘开

\frac{2 x ^{2} - 10 x + 14}{( x - 3 ) ^{3}} : \frac{2 x ^{2}}{( x - 3 ) ^{3}} - \frac{10 x}{( x - 3 ) ^{3}} + \frac{14}{( x - 3 ) ^{3}}

= \int \frac{2 x ^{2}}{( x - 3 ) ^{3}} - \frac{10 x}{( x - 3 ) ^{3}} + \frac{14}{( x - 3 ) ^{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 ^{2}}{( x - 3 ) ^{3}} d x - \int \frac{10 x}{( x - 3 ) ^{3}} d x + \int \frac{14}{( x - 3 ) ^{3}} d x

\int \frac{2 x ^{2}}{( x - 3 ) ^{3}} d x = 2 ln ∣ x - 3 ∣ - \frac{12}{x - 3} - \frac{9}{( x - 3 ) ^{2}}

\int \frac{10 x}{( x - 3 ) ^{3}} d x = 10 (- \frac{1}{x - 3} - \frac{3}{2 ( x - 3 ) ^{2}})

\int \frac{14}{( x - 3 ) ^{3}} d x = - \frac{7}{( x - 3 ) ^{2}}

= 2 ln ∣ x - 3 ∣ - \frac{12}{x - 3} - \frac{9}{( x - 3 ) ^{2}} - 10 (- \frac{1}{x - 3} - \frac{3}{2 ( x - 3 ) ^{2}}) - \frac{7}{( x - 3 ) ^{2}}

化简

2 ln ∣ x - 3 ∣ - \frac{12}{x - 3} - \frac{9}{( x - 3 ) ^{2}} - 10 (- \frac{1}{x - 3} - \frac{3}{2 ( x - 3 ) ^{2}}) - \frac{7}{( x - 3 ) ^{2}} : 2 ln ∣ x - 3 ∣ - \frac{2}{x - 3} - \frac{1}{( x - 3 ) ^{2}}

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

解答补常数

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

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