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人気のある微分積分 >

integral from 0 to 1 of 1/(x^2-3)

解

\int_{0}^{1} \frac{1}{x ^{2} - 3} d x

解

- \frac{ln ( \frac{1}{3} + 1 ) - ln ( - \frac{1}{3} + 1 )}{2 3}

+1

十進法表記

- 0.38017 \dots

解答ステップ

\int_{0}^{1} \frac{1}{x ^{2} - 3} d x

置換積分法を適用する

= \int_{0}^{\frac{1}{3}} \frac{1}{3 ( u ^{2} - 1 )} d u

定数を除く:

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

= \frac{1}{3} \cdot \int_{0}^{\frac{1}{3}} \frac{1}{u ^{2} - 1} d u

因数

u^{2} - 1 : - (- u^{2} + 1)

= \frac{1}{3} \cdot \int_{0}^{\frac{1}{3}} \frac{1}{- ( - u ^{2} + 1 )} d u

定数を除く:

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

= \frac{1}{3} (- \int_{0}^{\frac{1}{3}} \frac{1}{- u ^{2} + 1} d u)

共通積分を使用する:

\int \frac{1}{- u ^{2} + 1} d u = \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

= \frac{1}{3} (- [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{3}})

簡素化

\frac{1}{3} (- [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{3}}) : - \frac{1}{3} [\frac{ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{3}}

= - \frac{1}{3} [\frac{ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{3}}

限度を計算する:

\frac{ln ( \frac{1}{3} + 1 ) - ln ( - \frac{1}{3} + 1 )}{2}

= - \frac{1}{3} \cdot \frac{ln ( \frac{1}{3} + 1 ) - ln ( - \frac{1}{3} + 1 )}{2}

簡素化

= - \frac{ln ( \frac{1}{3} + 1 ) - ln ( - \frac{1}{3} + 1 )}{2 3}

グラフ

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