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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

对

x^{2} + 4 x + 2

配方:

(x + 2)^{2} - 2

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

使用换元积分法

= \int \frac{1}{u ^{2} - 2} 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 (arcsec (\frac{1}{2} (x + 2))) + sec (arcsec (\frac{1}{2} (x + 2)))

化简

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

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

解答补常数

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

作图

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