integral from-2 to 2 of (4-x^2)^{3/2}

解

\int_{- 2}^{2} (4 - x^{2})^{\frac{3}{2}} d x

6 π

十進法表記

18.84955 \dots

解答ステップ

\int_{- 2}^{2} (4 - x^{2})^{\frac{3}{2}} d x

部分積分を適用する

= [x (4 - x^{2})^{\frac{3}{2}} - \int - 3 x^{2} (4 - x^{2})^{\frac{1}{2}} d x]_{- 2}^{2}

\int - 3 x^{2} (4 - x^{2})^{\frac{1}{2}} d x = - 3 (\frac{1}{4} x^{3} (- x^{2} + 4)^{\frac{1}{2}} + 2 arcsin (\frac{1}{2} x) - sin (2 arcsin (\frac{1}{2} x)))

= [x (4 - x^{2})^{\frac{3}{2}} - (- 3 (\frac{1}{4} x^{3} (- x^{2} + 4)^{\frac{1}{2}} + 2 arcsin (\frac{1}{2} x) - sin (2 arcsin (\frac{1}{2} x))))]_{- 2}^{2}

簡素化

= [x (4 - x^{2})^{\frac{3}{2}} + 3 (\frac{1}{4} x^{3} (- x^{2} + 4)^{\frac{1}{2}} + 2 arcsin (\frac{1}{2} x) - sin (2 arcsin (\frac{1}{2} x)))]_{- 2}^{2}

限度を計算する:

6 π

= 6 π