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受欢迎的微积分 >

integral of (2x-8)/(sqrt(1-x-x^2))

解答

\int \frac{2 x - 8}{1 - x - x ^{2}} d x

解答

- 2 - x^{2} - x + 1 - 9 arcsin (\frac{1}{5} (2 x + 1)) + C

求解步骤

\int \frac{2 x - 8}{1 - x - x ^{2}} d x

对

1 - x - x^{2}

配方:

- (x + \frac{1}{2})^{2} + \frac{5}{4}

= \int \frac{2 x - 8}{- ( x + \frac{1}{2} ) ^{2} + \frac{5}{4}} d x

使用换元积分法

= \int \frac{2 ( 2 u - 9 )}{- 4 u ^{2} + 5} d u

提出常数:

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

= 2 \cdot \int \frac{2 u - 9}{- 4 u ^{2} + 5} d u

乘开

\frac{2 u - 9}{- 4 u ^{2} + 5} : \frac{2 u}{- 4 u ^{2} + 5} - \frac{9}{- 4 u ^{2} + 5}

= 2 \cdot \int \frac{2 u}{- 4 u ^{2} + 5} - \frac{9}{- 4 u ^{2} + 5} 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

= 2 (\int \frac{2 u}{- 4 u ^{2} + 5} d u - \int \frac{9}{- 4 u ^{2} + 5} d u)

\int \frac{2 u}{- 4 u ^{2} + 5} d u = - \frac{1}{2} - 4 u^{2} + 5

\int \frac{9}{- 4 u ^{2} + 5} d u = \frac{9}{2} arcsin (\frac{2}{5} u)

= 2 (- \frac{1}{2} - 4 u^{2} + 5 - \frac{9}{2} arcsin (\frac{2}{5} u))

u = x + \frac{1}{2}

代回

= 2 - \frac{1}{2} - 4 (x + \frac{1}{2})^{2} + 5 - \frac{9}{2} arcsin (\frac{2}{5} (x + \frac{1}{2}))

化简

2 - \frac{1}{2} - 4 (x + \frac{1}{2})^{2} + 5 - \frac{9}{2} arcsin (\frac{2}{5} (x + \frac{1}{2})) : - 2 - x^{2} - x + 1 - 9 arcsin (\frac{1}{5} (2 x + 1))

= - 2 - x^{2} - x + 1 - 9 arcsin (\frac{1}{5} (2 x + 1))

解答补常数

= - 2 - x^{2} - x + 1 - 9 arcsin (\frac{1}{5} (2 x + 1)) + C

作图

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