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

解答

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

- \frac{1}{3} (1 - x^{2})^{\frac{3}{2}} + 1 - x^{2} + C

求解步骤

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

提出常数:

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

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

使用换元积分法

= - \int u^{2} - 1 d u

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 u^{2} d u - \int 1 d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int 1 d u = u

= - (\frac{u ^{3}}{3} - u)

u = 1 - x^{2}

代回

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

化简

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

= - \frac{1}{3} (1 - x^{2})^{\frac{3}{2}} + 1 - x^{2}

解答补常数

= - \frac{1}{3} (1 - x^{2})^{\frac{3}{2}} + 1 - x^{2} + C

\frac{d}{d x} (e^{a x^{5}})

x \to 0 lim (2 x (1 - ln (x)))

\frac{d}{d x} (arcsin (x^{9}))

d er i v a t i v e f (x) = 6 x^{2} - 4 x

\frac{\partial}{\partial x} (x (5 z^{4} - 4 y^{3}))