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受欢迎的微积分 >

integral of x^3sqrt(a^2+x^2)

解答

\int x^{3} a^{2} + x^{2} d x

解答

\frac{1}{15} (3 x^{2} - 2 a^{2}) (x^{2} + a^{2})^{\frac{3}{2}} + C

求解步骤

\int x^{3} a^{2} + x^{2} d x

化简

x^{3} : x^{2} x

= \int x^{2} x a^{2} + x^{2} d x

使用换元积分法

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

使用换元积分法

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

乘开

(v - a^{2}) v : v^{\frac{3}{2}} - a^{2} v

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

\int v^{\frac{3}{2}} d v = \frac{2}{5} v^{\frac{5}{2}}

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

= \frac{1}{2} (\frac{2}{5} v^{\frac{5}{2}} - a^{2} \frac{2}{3} v^{\frac{3}{2}})

代回

= \frac{1}{2} (\frac{2}{5} (a^{2} + x^{2})^{\frac{5}{2}} - a^{2} \frac{2}{3} (a^{2} + x^{2})^{\frac{3}{2}})

化简

\frac{1}{2} (\frac{2}{5} (a^{2} + x^{2})^{\frac{5}{2}} - a^{2} \frac{2}{3} (a^{2} + x^{2})^{\frac{3}{2}}) : \frac{1}{15} (3 x^{2} - 2 a^{2}) (x^{2} + a^{2})^{\frac{3}{2}}

= \frac{1}{15} (3 x^{2} - 2 a^{2}) (x^{2} + a^{2})^{\frac{3}{2}}

解答补常数

= \frac{1}{15} (3 x^{2} - 2 a^{2}) (x^{2} + a^{2})^{\frac{3}{2}} + C

作图

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