integral of (x^3+2)^2

解答

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

\frac{x ^{7}}{7} + x^{4} + 4 x + C

求解步骤

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

乘开

(x^{3} + 2)^{2} : x^{6} + 4 x^{3} + 4

= \int x^{6} + 4 x^{3} + 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 x^{6} d x + \int 4 x^{3} d x + \int 4 d x

\int x^{6} d x = \frac{x ^{7}}{7}

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

\int 4 d x = 4 x

= \frac{x ^{7}}{7} + x^{4} + 4 x

解答补常数

= \frac{x ^{7}}{7} + x^{4} + 4 x + C

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