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受欢迎的微积分 >

integral of ((8x-3)/(sqrt(12x-4x^2-5)))

解答

\int (\frac{8 x - 3}{12 x - 4 x ^{2} - 5}) d x

解答

\frac{1}{2} (- 4 - 4 x^{2} + 12 x - 5 + 9 arcsin (\frac{2 x - 3}{2})) + C

求解步骤

\int \frac{8 x - 3}{12 x - 4 x ^{2} - 5} d x

使用换元积分法

= \int \frac{u}{2 - u ^{2} + 18 u - 17} d u

提出常数:

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

= \frac{1}{2} \cdot \int \frac{u}{- u ^{2} + 18 u - 17} d u

对

- u^{2} + 18 u - 17

配方:

- (u - 9)^{2} + 64

= \frac{1}{2} \cdot \int \frac{u}{- ( u - 9 ) ^{2} + 64} d u

使用换元积分法

= \frac{1}{2} \cdot \int \frac{v + 9}{- v ^{2} + 64} d v

使用三角换元法

= \frac{1}{2} \cdot \int 8 sin (w) + 9 d w

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

= \frac{1}{2} (\int 8 sin (w) d w + \int 9 d w)

\int 8 sin (w) d w = - 8 cos (w)

\int 9 d w = 9 w

= \frac{1}{2} (- 8 cos (w) + 9 w)

代回

= \frac{1}{2} (- 8 cos (arcsin (\frac{1}{8} (8 x - 3 - 9))) + 9 arcsin (\frac{1}{8} (8 x - 3 - 9)))

化简

\frac{1}{2} (- 8 cos (arcsin (\frac{1}{8} (8 x - 3 - 9))) + 9 arcsin (\frac{1}{8} (8 x - 3 - 9))) : \frac{1}{2} (- 4 - 4 x^{2} + 12 x - 5 + 9 arcsin (\frac{2 x - 3}{2}))

= \frac{1}{2} (- 4 - 4 x^{2} + 12 x - 5 + 9 arcsin (\frac{2 x - 3}{2}))

解答补常数

= \frac{1}{2} (- 4 - 4 x^{2} + 12 x - 5 + 9 arcsin (\frac{2 x - 3}{2})) + C

作图

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