integral of sqrt(9x^{-6)-81x^{-8}}

解答

\int 9 x^{- 6} - 81 x^{- 8} d x

3 \frac{x ^{2} - 9}{x ^{3}} - 3 - \frac{x ^{2} - 9}{9 x} + \frac{4}{27} \frac{x ^{2} - 9}{x} - \frac{( x ^{2} - 9 ) ^{\frac{3}{2}}}{3 x ^{3}} + C

求解步骤

\int 9 x^{- 6} - 81 x^{- 8} d x

9 x^{- 6} - 81 x^{- 8} = 9 \frac{1}{x ^{6}} - \frac{9}{x ^{8}}

= \int 9 \frac{1}{x ^{6}} - \frac{9}{x ^{8}} d x

提出常数:

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

= 9 \cdot \int \frac{1}{x ^{6}} - \frac{9}{x ^{8}} d x

9 = 3

= 3 \cdot \int \frac{1}{x ^{6}} - \frac{9}{x ^{8}} d x

使用分布积分法

= 3 (\frac{x ^{2} - 9}{x ^{8}} x - \int \frac{3 x ^{8} ( - x ^{2} + 12 )}{x ^{8} x ^{2} - 9} d x)

\int \frac{3 x ^{8} ( - x ^{2} + 12 )}{x ^{8} x ^{2} - 9} d x = 3 - \frac{x ^{2} - 9}{9 x} + \frac{4}{27} \frac{x ^{2} - 9}{x} - \frac{( x ^{2} - 9 ) ^{\frac{3}{2}}}{3 x ^{3}}

= 3 \frac{x ^{2} - 9}{x ^{8}} x - 3 - \frac{x ^{2} - 9}{9 x} + \frac{4}{27} \frac{x ^{2} - 9}{x} - \frac{( x ^{2} - 9 ) ^{\frac{3}{2}}}{3 x ^{3}}

\frac{x ^{2} - 9}{x ^{8}} x = \frac{x ^{2} - 9}{x ^{3}}

= 3 \frac{x ^{2} - 9}{x ^{3}} - 3 - \frac{x ^{2} - 9}{9 x} + \frac{4}{27} \frac{x ^{2} - 9}{x} - \frac{( x ^{2} - 9 ) ^{\frac{3}{2}}}{3 x ^{3}}

解答补常数

= 3 \frac{x ^{2} - 9}{x ^{3}} - 3 - \frac{x ^{2} - 9}{9 x} + \frac{4}{27} \frac{x ^{2} - 9}{x} - \frac{( x ^{2} - 9 ) ^{\frac{3}{2}}}{3 x ^{3}} + C

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

\int \frac{4 x ^{2}}{( 1 - 2 x ^{3} ) ^{4}} d x

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

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

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