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

解

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

2 - \frac{∣ θ ∣}{θ}

解答ステップ

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

定数を除く:

\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

総和規則を適用する:

\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 ∣)

簡素化

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

= 2 - \frac{∣ θ ∣}{θ}