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

解答

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

\frac{x ^{6}}{3} - x^{4} + x^{2} + C

求解步骤

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

提出常数:

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

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

乘开

x (x^{2} - 1)^{2} : x^{5} - 2 x^{3} + x

= 2 \cdot \int x^{5} - 2 x^{3} + x 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

= 2 (\int x^{5} d x - \int 2 x^{3} d x + \int x d x)

\int x^{5} d x = \frac{x ^{6}}{6}

\int 2 x^{3} d x = \frac{x ^{4}}{2}

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

= 2 (\frac{x ^{6}}{6} - \frac{x ^{4}}{2} + \frac{x ^{2}}{2})

化简

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

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

解答补常数

= \frac{x ^{6}}{3} - x^{4} + x^{2} + C

a re a x = 20 - 5 y^{2}, x = 5 y^{2} - 20

x \to 3 lim (\frac{3}{9 - x ^{2}})

a re a y = x 4 - x^{2}, (2, - 2)

d er i v a t i v e 8 t - 8 e^{t}

\frac{d}{d x} ((cos (4 x))^{x})