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

解答

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

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

求解步骤

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

使用分布积分法

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

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

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

化简

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

解答补常数

= \frac{x ^{2}}{( x ^{2} + a ^{2} ) ^{\frac{1}{2}}} + \frac{a ^{2}}{( x ^{2} + a ^{2} ) ^{\frac{1}{2}}} + C

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