integral from 0 to a of 4xsqrt(a^2-x^2)

解

\int_{0}^{a} 4 x a^{2} - x^{2} d x

\frac{4 ( a ^{2} ) ^{\frac{3}{2}}}{3}

解答ステップ

\int_{0}^{a} 4 x a^{2} - x^{2} d x

定数を除く:

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

= 4 \cdot \int_{0}^{a} x a^{2} - x^{2} d x

u置換積分法を適用する

= 4 \cdot \int_{a^{2}}^{0} - \frac{u}{2} d u

定数を除く:

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

= 4 (- \frac{1}{2} \cdot \int_{a^{2}}^{0} u d u)

べき乗則を適用する

= 4 (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0})

簡素化

4 (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0}) : - 2 [\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0}

= - 2 [\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0}

限度を計算する:

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

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

簡素化

= \frac{4 ( a ^{2} ) ^{\frac{3}{2}}}{3}