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

解答

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

\frac{2}{7} x^{\frac{7}{2}} - \frac{4}{3} x^{\frac{3}{2}} + C

求解步骤

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

乘开

x (x^{2} - 2) : x^{\frac{5}{2}} - 2 x

= \int x^{\frac{5}{2}} - 2 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^{\frac{5}{2}} d x - \int 2 x d x

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

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

= \frac{2}{7} x^{\frac{7}{2}} - \frac{4}{3} x^{\frac{3}{2}}

解答补常数

= \frac{2}{7} x^{\frac{7}{2}} - \frac{4}{3} x^{\frac{3}{2}} + C

d er i v a t i v e \frac{1}{4 x ^{2} + 5 x}

x \to \infty lim (3 x^{2} \cdot e^{- x})

t an g e n t f (x) = 8 x^{2} - x^{3}, (1, 7)

t a y l or (x - 5)^{\frac{7}{3}}

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