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人気のある微分積分 >

integral of (16)/(16-x^2)

解

\int \frac{16}{16 - x ^{2}} d x

解

4 (\frac{1}{2} ln \frac{x}{4} + 1 - \frac{1}{2} ln \frac{x}{4} - 1) + C

解答ステップ

\int \frac{16}{16 - x ^{2}} d x

定数を除く:

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

= 16 \cdot \int \frac{1}{16 - x ^{2}} d x

置換積分法を適用する

= 16 \cdot \int \frac{1}{4 ( - u ^{2} + 1 )} d u

定数を除く:

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

= 16 \cdot \frac{1}{4} \cdot \int \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}

= 16 \cdot \frac{1}{4} (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2})

代用を戻す

u = \frac{x}{4}

= 16 \cdot \frac{1}{4} (\frac{ln \frac{x}{4} + 1}{2} - \frac{ln \frac{x}{4} - 1}{2})

簡素化

16 \cdot \frac{1}{4} (\frac{ln \frac{x}{4} + 1}{2} - \frac{ln \frac{x}{4} - 1}{2}) : 4 (\frac{1}{2} ln \frac{x}{4} + 1 - \frac{1}{2} ln \frac{x}{4} - 1)

= 4 (\frac{1}{2} ln \frac{x}{4} + 1 - \frac{1}{2} ln \frac{x}{4} - 1)

定数を解答に追加する

= 4 (\frac{1}{2} ln \frac{x}{4} + 1 - \frac{1}{2} ln \frac{x}{4} - 1) + C

グラフ

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