integral from-k to k of |kx|

解答

\int_{- k}^{k} ∣ k x ∣ d x

k ∣ k ∣^{2}

求解步骤

\int_{- k}^{k} ∣ k x ∣ d x

使用绝对值运算法则:

∣ a \cdot b ∣ = ∣ a ∣ ∣ b ∣

= \int_{- k}^{k} ∣ k ∣ ∣ x ∣ d x

提出常数:

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

= ∣ k ∣ \cdot \int_{- k}^{k} ∣ x ∣ d x

使用积分常见定则:

\int ∣ x ∣ d x = \frac{x ∣ x ∣}{2}

= ∣ k ∣ [\frac{x ∣ x ∣}{2}]_{- k}^{k}

带入上下限计算后得:

k ∣ k ∣

= ∣ k ∣ k ∣ k ∣

化简

= k ∣ k ∣^{2}

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