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受欢迎的微积分 >

integral of (x^2-4x+4)/(sqrt(x^2-4x+53))

解答

\int \frac{x ^{2} - 4 x + 4}{x ^{2} - 4 x + 53} d x

解答

49 (- ln (\frac{1}{7} x - 2 + x^{2} - 4 x + 53) + \frac{1}{98} (x - 2) x^{2} - 4 x + 53 + \frac{1}{2} ln (\frac{x - 2 + x ^{2} - 4 x + 53}{7})) + C

求解步骤

\int \frac{x ^{2} - 4 x + 4}{x ^{2} - 4 x + 53} d x

对

x^{2} - 4 x + 4

配方:

(x - 2)^{2}

= \int \frac{( x - 2 ) ^{2}}{x ^{2} - 4 x + 53} d x

使用换元积分法

= \int \frac{u ^{2}}{u ^{2} + 49} d u

使用三角换元法

= \int 49 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

= 49 \cdot \int tan^{2} (v) sec (v) d v

使用三角恒等式改写

= 49 \cdot \int (- 1 + sec^{2} (v)) sec (v) d v

乘开

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

= 49 \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

= 49 (- \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) ∣

= 49 (- ln ∣ tan (v) + sec (v) ∣ + \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

代回

= 49 (- ln tan (arctan (\frac{1}{7} (x - 2))) + sec (arctan (\frac{1}{7} (x - 2))) + \frac{1}{2} sec (arctan (\frac{1}{7} (x - 2))) tan (arctan (\frac{1}{7} (x - 2))) + \frac{1}{2} ln tan (arctan (\frac{1}{7} (x - 2))) + sec (arctan (\frac{1}{7} (x - 2))))

化简

49 (- ln tan (arctan (\frac{1}{7} (x - 2))) + sec (arctan (\frac{1}{7} (x - 2))) + \frac{1}{2} sec (arctan (\frac{1}{7} (x - 2))) tan (arctan (\frac{1}{7} (x - 2))) + \frac{1}{2} ln tan (arctan (\frac{1}{7} (x - 2))) + sec (arctan (\frac{1}{7} (x - 2)))) : 49 (- ln (\frac{1}{7} x - 2 + x^{2} - 4 x + 53) + \frac{1}{98} (x - 2) x^{2} - 4 x + 53 + \frac{1}{2} ln (\frac{x - 2 + x ^{2} - 4 x + 53}{7}))

= 49 (- ln (\frac{1}{7} x - 2 + x^{2} - 4 x + 53) + \frac{1}{98} (x - 2) x^{2} - 4 x + 53 + \frac{1}{2} ln (\frac{x - 2 + x ^{2} - 4 x + 53}{7}))

解答补常数

= 49 (- ln (\frac{1}{7} x - 2 + x^{2} - 4 x + 53) + \frac{1}{98} (x - 2) x^{2} - 4 x + 53 + \frac{1}{2} ln (\frac{x - 2 + x ^{2} - 4 x + 53}{7})) + C

作图

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