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受欢迎的微积分 >

integral of x/(sqrt(2+x-x^2))

解答

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

解答

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

求解步骤

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

对

2 + x - x^{2}

配方:

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

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

使用换元积分法

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

乘开

\frac{2 u + 1}{- 4 u ^{2} + 9} : \frac{2 u}{- 4 u ^{2} + 9} + \frac{1}{- 4 u ^{2} + 9}

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

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

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

= - \frac{1}{2} - 4 u^{2} + 9 + \frac{1}{2} arcsin (\frac{2}{3} u)

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

代回

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

化简

- \frac{1}{2} - 4 (x - \frac{1}{2})^{2} + 9 + \frac{1}{2} arcsin (\frac{2}{3} (x - \frac{1}{2})) : - - x^{2} + x + 2 + \frac{1}{2} arcsin (\frac{2}{3} x - \frac{1}{3})

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

解答补常数

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

作图

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