integral of ((x^3)/3-(x^4)/2+(x^5)/5)

解答

\int (\frac{x ^{3}}{3} - \frac{x ^{4}}{2} + \frac{x ^{5}}{5}) d x

\frac{x ^{4}}{12} - \frac{x ^{5}}{10} + \frac{x ^{6}}{30} + C

求解步骤

\int \frac{x ^{3}}{3} - \frac{x ^{4}}{2} + \frac{x ^{5}}{5} 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 ^{3}}{3} d x - \int \frac{x ^{4}}{2} d x + \int \frac{x ^{5}}{5} d x

\int \frac{x ^{3}}{3} d x = \frac{x ^{4}}{12}

\int \frac{x ^{4}}{2} d x = \frac{x ^{5}}{10}

\int \frac{x ^{5}}{5} d x = \frac{x ^{6}}{30}

= \frac{x ^{4}}{12} - \frac{x ^{5}}{10} + \frac{x ^{6}}{30}

解答补常数

= \frac{x ^{4}}{12} - \frac{x ^{5}}{10} + \frac{x ^{6}}{30} + C

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