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受欢迎的微积分 >

integral of (x^3)/((a^2+x^2)^{3/2)}

解答

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

解答

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

求解步骤

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

使用换元积分法

= \int \frac{u}{2 ( u + a ^{2} ) ^{\frac{3}{2}}} d u

提出常数:

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

= \frac{1}{2} \cdot \int \frac{u}{( u + a ^{2} ) ^{\frac{3}{2}}} d u

使用换元积分法

= \frac{1}{2} \cdot \int \frac{v - a ^{2}}{v ^{\frac{3}{2}}} d v

乘开

\frac{v - a ^{2}}{v ^{\frac{3}{2}}} : \frac{1}{v ^{\frac{1}{2}}} - \frac{a ^{2}}{v ^{\frac{3}{2}}}

= \frac{1}{2} \cdot \int \frac{1}{v ^{\frac{1}{2}}} - \frac{a ^{2}}{v ^{\frac{3}{2}}} d v

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{v ^{\frac{1}{2}}} d v = 2 v^{\frac{1}{2}}

\int \frac{a ^{2}}{v ^{\frac{3}{2}}} d v = - \frac{2 a ^{2}}{v ^{\frac{1}{2}}}

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

代回

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

化简

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

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

解答补常数

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

作图

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