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受欢迎的微积分 >

integral of-1/(9xsqrt(81x^2-9))

解答

\int - \frac{1}{9 x 81 x ^{2} - 9} d x

解答

- \frac{1}{27} arcsec (3 x) + C

求解步骤

\int - \frac{1}{9 x 81 x ^{2} - 9} d x

提出常数:

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

= - \frac{1}{9} \cdot \int \frac{1}{x 81 x ^{2} - 9} d x

使用三角换元法

= - \frac{1}{9} \cdot \int \frac{1}{3} d u

提出常数:

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

= - \frac{1}{9} \cdot \frac{1}{3} \cdot \int 1 d u

常数积分:

\int a d x = a x

= - \frac{1}{9} \cdot \frac{1}{3} \cdot 1 \cdot u

u = arcsec (\frac{9}{3} x)

代回

= - \frac{1}{9} \cdot \frac{1}{3} \cdot 1 \cdot arcsec (\frac{9}{3} x)

化简

- \frac{1}{9} \cdot \frac{1}{3} \cdot 1 \cdot arcsec (\frac{9}{3} x) : - \frac{1}{27} arcsec (3 x)

= - \frac{1}{27} arcsec (3 x)

解答补常数

= - \frac{1}{27} arcsec (3 x) + C

作图

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