integral of 1/3 (1-x^2)^3

解答

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

\frac{1}{3} x - \frac{1}{3} x^{3} + \frac{x ^{5}}{5} - \frac{x ^{7}}{21} + C

求解步骤

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

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int (1 - x^{2})^{3} d x

乘开

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

= \frac{1}{3} \cdot \int 1 - 3 x^{2} + 3 x^{4} - x^{6} 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

= \frac{1}{3} (\int 1 d x - \int 3 x^{2} d x + \int 3 x^{4} d x - \int x^{6} d x)

\int 1 d x = x

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

\int 3 x^{4} d x = \frac{3 x ^{5}}{5}

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

= \frac{1}{3} (x - x^{3} + \frac{3 x ^{5}}{5} - \frac{x ^{7}}{7})

化简

\frac{1}{3} (x - x^{3} + \frac{3 x ^{5}}{5} - \frac{x ^{7}}{7}) : \frac{1}{3} x - \frac{1}{3} x^{3} + \frac{x ^{5}}{5} - \frac{x ^{7}}{21}

= \frac{1}{3} x - \frac{1}{3} x^{3} + \frac{x ^{5}}{5} - \frac{x ^{7}}{21}

解答补常数

= \frac{1}{3} x - \frac{1}{3} x^{3} + \frac{x ^{5}}{5} - \frac{x ^{7}}{21} + C

cos (x) \frac{d y}{d x} + (sin (x)) y = 1

x \to 3 lim (x^{3})

\int \frac{x + 2}{2 x + 2} d x

\frac{\partial}{\partial y} (4 x + 9 y - 360)

d er i v a t i v e \frac{( 2 x + 1 ) ( x ^{3} + 3 )}{x}