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受欢迎的微积分 >

integral of (x^2-6x+9)/(sqrt(x^2-6x+18))

解答

\int \frac{x ^{2} - 6 x + 9}{x ^{2} - 6 x + 18} d x

解答

9 (- ln (\frac{1}{3} x - 3 + x^{2} - 6 x + 18) + \frac{1}{18} (x - 3) x^{2} - 6 x + 18 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 18}{3})) + C

求解步骤

\int \frac{x ^{2} - 6 x + 9}{x ^{2} - 6 x + 18} d x

对

x^{2} - 6 x + 9

配方:

(x - 3)^{2}

= \int \frac{( x - 3 ) ^{2}}{x ^{2} - 6 x + 18} d x

使用换元积分法

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

使用三角换元法

= \int 9 tan^{2} (v) sec (v) d v

提出常数:

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

= 9 \cdot \int tan^{2} (v) sec (v) d v

使用三角恒等式改写

= 9 \cdot \int (- 1 + sec^{2} (v)) sec (v) d v

乘开

(- 1 + sec^{2} (v)) sec (v) : - sec (v) + sec^{3} (v)

= 9 \cdot \int - sec (v) + sec^{3} (v) 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

= 9 (- \int sec (v) d v + \int sec^{3} (v) d v)

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

\int sec^{3} (v) d v = \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣

= 9 (- ln ∣ tan (v) + sec (v) ∣ + \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

代回

= 9 (- ln tan (arctan (\frac{1}{3} (x - 3))) + sec (arctan (\frac{1}{3} (x - 3))) + \frac{1}{2} sec (arctan (\frac{1}{3} (x - 3))) tan (arctan (\frac{1}{3} (x - 3))) + \frac{1}{2} ln tan (arctan (\frac{1}{3} (x - 3))) + sec (arctan (\frac{1}{3} (x - 3))))

化简

9 (- ln tan (arctan (\frac{1}{3} (x - 3))) + sec (arctan (\frac{1}{3} (x - 3))) + \frac{1}{2} sec (arctan (\frac{1}{3} (x - 3))) tan (arctan (\frac{1}{3} (x - 3))) + \frac{1}{2} ln tan (arctan (\frac{1}{3} (x - 3))) + sec (arctan (\frac{1}{3} (x - 3)))) : 9 (- ln (\frac{1}{3} x - 3 + x^{2} - 6 x + 18) + \frac{1}{18} (x - 3) x^{2} - 6 x + 18 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 18}{3}))

= 9 (- ln (\frac{1}{3} x - 3 + x^{2} - 6 x + 18) + \frac{1}{18} (x - 3) x^{2} - 6 x + 18 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 18}{3}))

解答补常数

= 9 (- ln (\frac{1}{3} x - 3 + x^{2} - 6 x + 18) + \frac{1}{18} (x - 3) x^{2} - 6 x + 18 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 18}{3})) + C

作图

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