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

解答

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

3 (2 π^{2} - 16)

十进制

11.21762 \dots

求解步骤

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

提出常数:

\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

使用分布积分法

= 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}

带入上下限计算后得:

2 π^{2} - 16

= 3 (2 π^{2} - 16)

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