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

解

\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

解答ステップ

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

部分積分を適用する

= \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}}})

簡素化

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

定数を解答に追加する

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