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受欢迎的微积分 >

integral of 5/(sqrt(25x^2-9))

解答

\int \frac{5}{25 x ^{2} - 9} d x

解答

ln (\frac{1}{3} 25 x^{2} - 9 + 5 x) + C

求解步骤

\int \frac{5}{25 x ^{2} - 9} d x

提出常数:

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

= 5 \cdot \int \frac{1}{25 x ^{2} - 9} d x

使用三角换元法

= 5 \cdot \int \frac{sec ( u )}{5} d u

提出常数:

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

= 5 \cdot \frac{1}{5} \cdot \int sec (u) d u

使用积分常见定则:

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

= 5 \cdot \frac{1}{5} ln ∣ tan (u) + sec (u) ∣

u = arcsec (\frac{5}{3} x)

代回

= 5 \cdot \frac{1}{5} ln tan (arcsec (\frac{5}{3} x)) + sec (arcsec (\frac{5}{3} x))

化简

5 \cdot \frac{1}{5} ln tan (arcsec (\frac{5}{3} x)) + sec (arcsec (\frac{5}{3} x)) : ln (\frac{1}{3} 25 x^{2} - 9 + 5 x)

= ln (\frac{1}{3} 25 x^{2} - 9 + 5 x)

解答补常数

= ln (\frac{1}{3} 25 x^{2} - 9 + 5 x) + C

作图

流行的例子

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