integral from 0 to a of x^5sqrt(a^3-x^3)

解答

\int_{0}^{a} x^{5} a^{3} - x^{3} d x

\frac{4 a ^{\frac{15}{2}}}{45}

求解步骤

\int_{0}^{a} x^{5} a^{3} - x^{3} d x

使用换元积分法

= \int_{a^{3}}^{0} - \frac{( - u + a ^{3} ) u}{3} d u

提出常数:

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

= - \frac{1}{3} \cdot \int_{a^{3}}^{0} (- u + a^{3}) u d u

乘开

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

= - \frac{1}{3} \cdot \int_{a^{3}}^{0} - u^{\frac{3}{2}} + a^{3} u d u

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}{3} (- \int_{a^{3}}^{0} u^{\frac{3}{2}} d u + \int_{a^{3}}^{0} a^{3} u d u)

\int_{a^{3}}^{0} u^{\frac{3}{2}} d u = - \frac{2 ( a ^{3} ) ^{\frac{5}{2}}}{5}

\int_{a^{3}}^{0} a^{3} u d u = a^{3} (- \frac{2}{3} (a^{3})^{\frac{3}{2}})

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

化简

- \frac{1}{3} - - \frac{2 ( a ^{3} ) ^{\frac{5}{2}}}{5} + a^{3} (- \frac{2}{3} (a^{3})^{\frac{3}{2}}) : \frac{4 a ^{\frac{15}{2}}}{45}

= \frac{4 a ^{\frac{15}{2}}}{45}

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