integral of (2x^3+6x)^2

解答

\int (2 x^{3} + 6 x)^{2} d x

\frac{4 x ^{7}}{7} + \frac{24 x ^{5}}{5} + 12 x^{3} + C

求解步骤

\int (2 x^{3} + 6 x)^{2} d x

乘开

(2 x^{3} + 6 x)^{2} : 4 x^{6} + 24 x^{4} + 36 x^{2}

= \int 4 x^{6} + 24 x^{4} + 36 x^{2} 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 4 x^{6} d x + \int 24 x^{4} d x + \int 36 x^{2} d x

\int 4 x^{6} d x = \frac{4 x ^{7}}{7}

\int 24 x^{4} d x = \frac{24 x ^{5}}{5}

\int 36 x^{2} d x = 12 x^{3}

= \frac{4 x ^{7}}{7} + \frac{24 x ^{5}}{5} + 12 x^{3}

解答补常数

= \frac{4 x ^{7}}{7} + \frac{24 x ^{5}}{5} + 12 x^{3} + C

\frac{\partial}{\partial x} (5 ln (x y))

x^{^{''}} + 20 x^{^{'}} + 500 x = 12

\int_{0}^{4} x - x + 2 d x

\frac{d}{d x} (e^{x} + 2)

\int 6 e^{6 x} d x