integral of ((a^2-x^2)^{1/2})/(x^2)

解答

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

- \frac{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}}{x} - \frac{x}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} + \frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} - arcsin (\frac{x}{a}) + \frac{1}{2} sin (2 arcsin (\frac{x}{a})) + C

求解步骤

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

使用分布积分法

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

\int \frac{1}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} d x = \frac{x}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} - \frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} + arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a}))

= - \frac{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}}{x} - (\frac{x}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} - \frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} + arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a})))

化简

= - \frac{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}}{x} - \frac{x}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} + \frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} - arcsin (\frac{x}{a}) + \frac{1}{2} sin (2 arcsin (\frac{x}{a}))

解答补常数

= - \frac{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}}{x} - \frac{x}{( a ^{2} - x ^{2} ) ^{\frac{1}{2}}} + \frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} - arcsin (\frac{x}{a}) + \frac{1}{2} sin (2 arcsin (\frac{x}{a})) + C

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