integral of (2x^2+1)^3

解答

\int (2 x^{2} + 1)^{3} d x

\frac{8 x ^{7}}{7} + \frac{12 x ^{5}}{5} + 2 x^{3} + x + C

求解步骤

\int (2 x^{2} + 1)^{3} d x

乘开

(2 x^{2} + 1)^{3} : 8 x^{6} + 12 x^{4} + 6 x^{2} + 1

= \int 8 x^{6} + 12 x^{4} + 6 x^{2} + 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 8 x^{6} d x + \int 12 x^{4} d x + \int 6 x^{2} d x + \int 1 d x

\int 8 x^{6} d x = \frac{8 x ^{7}}{7}

\int 12 x^{4} d x = \frac{12 x ^{5}}{5}

\int 6 x^{2} d x = 2 x^{3}

\int 1 d x = x

= \frac{8 x ^{7}}{7} + \frac{12 x ^{5}}{5} + 2 x^{3} + x

解答补常数

= \frac{8 x ^{7}}{7} + \frac{12 x ^{5}}{5} + 2 x^{3} + x + C

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