integral of (((x^5-x^2)^2)/(sqrt(x)))

解答

\int (\frac{( x ^{5} - x ^{2} ) ^{2}}{x}) d x

\frac{2}{21} x^{\frac{21}{2}} - \frac{4}{15} x^{\frac{15}{2}} + \frac{2}{9} x^{\frac{9}{2}} + C

求解步骤

\int \frac{( x ^{5} - x ^{2} ) ^{2}}{x} d x

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

= \int \frac{( x ^{5} - x ^{2} ) ^{2} - x}{x} + 1 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{( x ^{5} - x ^{2} ) ^{2} - x}{x} d x + \int 1 d x

\int \frac{( x ^{5} - x ^{2} ) ^{2} - x}{x} d x = \frac{2}{21} x^{\frac{21}{2}} - \frac{4}{15} x^{\frac{15}{2}} + \frac{2}{9} x^{\frac{9}{2}} - x

\int 1 d x = x

= \frac{2}{21} x^{\frac{21}{2}} - \frac{4}{15} x^{\frac{15}{2}} + \frac{2}{9} x^{\frac{9}{2}} - x + x

化简

= \frac{2}{21} x^{\frac{21}{2}} - \frac{4}{15} x^{\frac{15}{2}} + \frac{2}{9} x^{\frac{9}{2}}

解答补常数

= \frac{2}{21} x^{\frac{21}{2}} - \frac{4}{15} x^{\frac{15}{2}} + \frac{2}{9} x^{\frac{9}{2}} + C

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