What is the integral from-θ to θ of 1/(θ^2)(θ-|x|) ?

The integral from-θ to θ of 1/(θ^2)(θ-|x|) is 2-(|θ|)/θ

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integral from-θ to θ of 1/(θ^2)(θ-|x|)

\int_{- θ}^{θ} \frac{1}{θ ^{2}} (θ - ∣ x ∣) d x

2 - \frac{∣ θ ∣}{θ}

Solution steps

\int_{- θ}^{θ} \frac{1}{θ ^{2}} (θ - ∣ x ∣) d x

Take the constant out:

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

= \frac{1}{θ ^{2}} \cdot \int_{- θ}^{θ} θ - ∣ x ∣ d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{θ ^{2}} (\int_{- θ}^{θ} θ d x - \int_{- θ}^{θ} ∣ x ∣ d x)

\int_{- θ}^{θ} θ d x = 2 θ^{2}

\int_{- θ}^{θ} ∣ x ∣ d x = θ ∣ θ ∣

= \frac{1}{θ ^{2}} (2 θ^{2} - θ ∣ θ ∣)

Simplify

\frac{1}{θ ^{2}} (2 θ^{2} - θ ∣ θ ∣) : 2 - \frac{∣ θ ∣}{θ}

= 2 - \frac{∣ θ ∣}{θ}

What is the integral from-θ to θ of 1/(θ^2)(θ-|x|) ?
The integral from-θ to θ of 1/(θ^2)(θ-|x|) is 2-(|θ|)/θ