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受欢迎的微积分 >

integral of (5-4x)/(sqrt(12x-4x^2-8))

解答

\int \frac{5 - 4 x}{12 x - 4 x ^{2} - 8} d x

解答

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

求解步骤

\int \frac{5 - 4 x}{12 x - 4 x ^{2} - 8} d x

对

12 x - 4 x^{2} - 8

配方:

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

= \int \frac{5 - 4 x}{- 4 ( x - \frac{3}{2} ) ^{2} + 1} d x

使用换元积分法

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

乘开

\frac{- 4 u - 1}{- 4 u ^{2} + 1} : - \frac{4 u}{- 4 u ^{2} + 1} - \frac{1}{- 4 u ^{2} + 1}

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

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

\int \frac{1}{- 4 u ^{2} + 1} d u = \frac{1}{2} arcsin (2 u)

= - (- - 4 u^{2} + 1) - \frac{1}{2} arcsin (2 u)

u = x - \frac{3}{2}

代回

= - - - 4 (x - \frac{3}{2})^{2} + 1 - \frac{1}{2} arcsin (2 (x - \frac{3}{2}))

化简

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

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

解答补常数

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

作图

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

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