integral from 0 to pi of 3x^2cos(x/2)

解

\int_{0}^{π} 3 x^{2} cos (\frac{x}{2}) d x

3 (2 π^{2} - 16)

十進法表記

11.21762 \dots

解答ステップ

\int_{0}^{π} 3 x^{2} cos (\frac{x}{2}) d x

定数を除く:

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

= 3 \cdot \int_{0}^{π} x^{2} cos (\frac{x}{2}) d x

部分積分を適用する

= 3 [2 x^{2} sin (\frac{x}{2}) - \int 4 x sin (\frac{x}{2}) d x]_{0}^{π}

\int 4 x sin (\frac{x}{2}) d x = 4 (- 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2}))

= 3 [2 x^{2} sin (\frac{x}{2}) - 4 (- 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2}))]_{0}^{π}

限度を計算する:

2 π^{2} - 16

= 3 (2 π^{2} - 16)