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受欢迎的微积分 >

integral of 1/(sqrt(12x-9x^2))

解答

\int \frac{1}{12 x - 9 x ^{2}} d x

解答

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

求解步骤

\int \frac{1}{12 x - 9 x ^{2}} d x

对

12 x - 9 x^{2}

配方:

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

= \int \frac{1}{- 9 ( x - \frac{2}{3} ) ^{2} + 4} d x

使用换元积分法

= \int \frac{1}{- 9 u ^{2} + 4} d u

使用三角换元法

= \int \frac{1}{3} d v

提出常数:

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

= \frac{1}{3} \cdot \int 1 d v

常数积分:

\int a d x = a x

= \frac{1}{3} \cdot 1 \cdot v

代回

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

化简

\frac{1}{3} \cdot 1 \cdot arcsin (\frac{3}{2} (x - \frac{2}{3})) : \frac{1}{3} arcsin (\frac{3}{2} x - 1)

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

解答补常数

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

作图

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