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受欢迎的微积分 >

integral of (3x-2)/(sqrt(3+2x-x^2))

解答

\int \frac{3 x - 2}{3 + 2 x - x ^{2}} d x

解答

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

求解步骤

\int \frac{3 x - 2}{3 + 2 x - x ^{2}} d x

使用换元积分法

= \int \frac{u}{- u ^{2} + 2 u + 35} d u

对

- u^{2} + 2 u + 35

配方:

- (u - 1)^{2} + 36

= \int \frac{u}{- ( u - 1 ) ^{2} + 36} d u

使用换元积分法

= \int \frac{v + 1}{- v ^{2} + 36} d v

乘开

\frac{v + 1}{- v ^{2} + 36} : \frac{v}{- v ^{2} + 36} + \frac{1}{- v ^{2} + 36}

= \int \frac{v}{- v ^{2} + 36} + \frac{1}{- v ^{2} + 36} d v

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{v}{- v ^{2} + 36} d v + \int \frac{1}{- v ^{2} + 36} d v

\int \frac{v}{- v ^{2} + 36} d v = - - v^{2} + 36

\int \frac{1}{- v ^{2} + 36} d v = arcsin (\frac{1}{6} v)

= - - v^{2} + 36 + arcsin (\frac{1}{6} v)

代回

= - - (3 x - 2 - 1)^{2} + 36 + arcsin (\frac{1}{6} (3 x - 2 - 1))

化简

- - (3 x - 2 - 1)^{2} + 36 + arcsin (\frac{1}{6} (3 x - 2 - 1)) : - 3 - x^{2} + 2 x + 3 + arcsin (\frac{1}{2} (x - 1))

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

解答补常数

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

作图

流行的例子

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