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受欢迎的微积分 >

integral of 1/(sqrt(x^2-4x+20))

解答

\int \frac{1}{x ^{2} - 4 x + 20} d x

解答

ln (\frac{1}{4} x - 2 + x^{2} - 4 x + 20) + C

求解步骤

\int \frac{1}{x ^{2} - 4 x + 20} d x

对

x^{2} - 4 x + 20

配方:

(x - 2)^{2} + 16

= \int \frac{1}{( x - 2 ) ^{2} + 16} d x

使用换元积分法

= \int \frac{1}{u ^{2} + 16} d u

使用三角换元法

= \int sec (v) d v

使用积分常见定则:

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

= ln ∣ tan (v) + sec (v) ∣

代回

= ln tan (arctan (\frac{1}{4} (x - 2))) + sec (arctan (\frac{1}{4} (x - 2)))

化简

ln tan (arctan (\frac{1}{4} (x - 2))) + sec (arctan (\frac{1}{4} (x - 2))) : ln (\frac{1}{4} x - 2 + x^{2} - 4 x + 20)

= ln (\frac{1}{4} x - 2 + x^{2} - 4 x + 20)

解答补常数

= ln (\frac{1}{4} x - 2 + x^{2} - 4 x + 20) + C

作图

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