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

解答

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

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

求解步骤

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

乘开

x (x^{2} + 2)^{2} : x^{5} + 4 x^{3} + 4 x

= \int x^{5} + 4 x^{3} + 4 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

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

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

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

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

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

解答补常数

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

in v erse l a pl a ce 4 (\frac{( 1 )}{s ^{2} + 16})

a re a y = \frac{2}{x}, y = 7 - x

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

(1 + x)^{2} \frac{d y}{d x} = (1 + y)^{2}

d er i v a t i v e x 9 - x