integral of 1/(x^2sqrt(1-4x^2))

解答

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

- \frac{1 - 4 x ^{2}}{x} + C

求解步骤

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

使用三角换元法

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

提出常数:

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

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

使用积分常见定则:

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

= 2 (- cot (u))

u = arcsin (2 x)

代回

= 2 (- cot (arcsin (2 x)))

化简

2 (- cot (arcsin (2 x))) : - \frac{1 - 4 x ^{2}}{x}

= - \frac{1 - 4 x ^{2}}{x}

解答补常数

= - \frac{1 - 4 x ^{2}}{x} + C

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