What is the integral from-k to k of |kx| ?

The integral from-k to k of |kx| is k|k|^2

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integral from-k to k of |kx|

\int_{- k}^{k} ∣ k x ∣ d x

k ∣ k ∣^{2}

Solution steps

\int_{- k}^{k} ∣ k x ∣ d x

Apply absolute rule:

∣ a \cdot b ∣ = ∣ a ∣ ∣ b ∣

= \int_{- k}^{k} ∣ k ∣ ∣ x ∣ d x

Take the constant out:

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

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

Use the common integral:

\int ∣ x ∣ d x = \frac{x ∣ x ∣}{2}

= ∣ k ∣ [\frac{x ∣ x ∣}{2}]_{- k}^{k}

Compute the boundaries:

k ∣ k ∣

= ∣ k ∣ k ∣ k ∣

Simplify

= k ∣ k ∣^{2}

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What is the integral from-k to k of |kx| ?
The integral from-k to k of |kx| is k|k|^2