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

解答

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

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

求解步骤

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

使用三角换元法

= \int \frac{cot ( u )}{cos ( u ) sin ( u )} d u

化简

\frac{cot ( u )}{cos ( u ) sin ( u )} : csc^{2} (u)

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

使用积分常见定则:

\int csc^{2} (u) d u = - cot (u)

= - cot (u)

u = arcsin (x)

代回

= - cot (arcsin (x))

化简

- cot (arcsin (x)) : - \frac{1 - x ^{2}}{x}

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

解答补常数

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

\int \frac{3 x ^{2} - 4 x + 1}{x ^{3} - 2 x ^{2} + x - 2} d x

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