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

解答

\int \frac{( x - 2 ) ^{2}}{x ^{4}} d x

- \frac{1}{x} + \frac{2}{x ^{2}} - \frac{4}{3 x ^{3}} + C

求解步骤

\int \frac{( x - 2 ) ^{2}}{x ^{4}} d x

乘开

\frac{( x - 2 ) ^{2}}{x ^{4}} : \frac{1}{x ^{2}} - \frac{4}{x ^{3}} + \frac{4}{x ^{4}}

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

\int \frac{1}{x ^{2}} d x = - \frac{1}{x}

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

\int \frac{4}{x ^{4}} d x = - \frac{4}{3 x ^{3}}

= - \frac{1}{x} - (- \frac{2}{x ^{2}}) - \frac{4}{3 x ^{3}}

化简

= - \frac{1}{x} + \frac{2}{x ^{2}} - \frac{4}{3 x ^{3}}

解答补常数

= - \frac{1}{x} + \frac{2}{x ^{2}} - \frac{4}{3 x ^{3}} + C

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