integral of (4x+3)(2x^2+3)^3

解答

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

4 x^{8} + 24 x^{6} + 54 x^{4} + 54 x^{2} + \frac{24 x ^{7}}{7} + \frac{108 x ^{5}}{5} + 54 x^{3} + 81 x + C

求解步骤

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

乘开

(4 x + 3) (2 x^{2} + 3)^{3} : 32 x^{7} + 144 x^{5} + 216 x^{3} + 108 x + 24 x^{6} + 108 x^{4} + 162 x^{2} + 81

= \int 32 x^{7} + 144 x^{5} + 216 x^{3} + 108 x + 24 x^{6} + 108 x^{4} + 162 x^{2} + 81 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 32 x^{7} d x + \int 144 x^{5} d x + \int 216 x^{3} d x + \int 108 x d x + \int 24 x^{6} d x + \int 108 x^{4} d x + \int 162 x^{2} d x + \int 81 d x

\int 32 x^{7} d x = 4 x^{8}

\int 144 x^{5} d x = 24 x^{6}

\int 216 x^{3} d x = 54 x^{4}

\int 108 x d x = 54 x^{2}

\int 24 x^{6} d x = \frac{24 x ^{7}}{7}

\int 108 x^{4} d x = \frac{108 x ^{5}}{5}

\int 162 x^{2} d x = 54 x^{3}

\int 81 d x = 81 x

= 4 x^{8} + 24 x^{6} + 54 x^{4} + 54 x^{2} + \frac{24 x ^{7}}{7} + \frac{108 x ^{5}}{5} + 54 x^{3} + 81 x

解答补常数

= 4 x^{8} + 24 x^{6} + 54 x^{4} + 54 x^{2} + \frac{24 x ^{7}}{7} + \frac{108 x ^{5}}{5} + 54 x^{3} + 81 x + C