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受欢迎的微积分 >

integral of 1/(sqrt(4x^2-25))

解答

\int \frac{1}{4 x ^{2} - 25} d x

解答

\frac{1}{2} ln (\frac{4 x ^{2} - 25 + 2 x}{5}) + C

求解步骤

\int \frac{1}{4 x ^{2} - 25} d x

使用三角换元法

= \int \frac{sec ( u )}{2} 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 sec (u) d u

使用积分常见定则:

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

= \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

u = arcsec (\frac{2}{5} x)

代回

= \frac{1}{2} ln tan (arcsec (\frac{2}{5} x)) + sec (arcsec (\frac{2}{5} x))

化简

\frac{1}{2} ln tan (arcsec (\frac{2}{5} x)) + sec (arcsec (\frac{2}{5} x)) : \frac{1}{2} ln (\frac{4 x ^{2} - 25 + 2 x}{5})

= \frac{1}{2} ln (\frac{4 x ^{2} - 25 + 2 x}{5})

解答补常数

= \frac{1}{2} ln (\frac{4 x ^{2} - 25 + 2 x}{5}) + C

作图

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