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

解答

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

\frac{x}{1 - x ^{2}} + C

求解步骤

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

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

假定

(1 - x^{2}) \geq 0

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

使用三角换元法

= \int sec^{2} (u) d u

使用积分常见定则:

\int sec^{2} (u) d u = tan (u)

= tan (u)

u = arcsin (x)

代回

= tan (arcsin (x))

利用以下特性:

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

= \frac{x}{1 - x ^{2}}

解答补常数

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

\frac{d}{d x} (\frac{ln ( x )}{9 x})

\int (\frac{1}{e ^{3 t}}) d t

x \to 0 lim (\frac{e - 1}{x})

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

x \to 1 lim (\frac{∣ 3 - x ∣}{3 - x})