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人気のある微分積分 >

integral of (x^2-6)/(x^2-9)

解

\int \frac{x ^{2} - 6}{x ^{2} - 9} d x

解

x - \frac{1}{2} ln \frac{x}{3} + 1 + \frac{1}{2} ln \frac{x}{3} - 1 + C

解答ステップ

\int \frac{x ^{2} - 6}{x ^{2} - 9} d x

置換積分法を適用する

= \int \frac{2 6 ( u ^{2} - 1 )}{2 u ^{2} - 3} d u

定数を除く:

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

= 26 \cdot \int \frac{u ^{2} - 1}{2 u ^{2} - 3} d u

長除法

\frac{u ^{2} - 1}{2 u ^{2} - 3} : \frac{1}{2} + \frac{1}{2 ( 2 u ^{2} - 3 )}

= 26 \cdot \int \frac{1}{2} + \frac{1}{2 ( 2 u ^{2} - 3 )} d u

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

= 26 (\int \frac{1}{2} d u + \int \frac{1}{2 ( 2 u ^{2} - 3 )} d u)

\int \frac{1}{2} d u = \frac{1}{2} u

\int \frac{1}{2 ( 2 u ^{2} - 3 )} d u = - \frac{1}{4 6} (ln \frac{2}{3} u + 1 - ln \frac{2}{3} u - 1)

= 26 (\frac{1}{2} u - \frac{1}{4 6} (ln \frac{2}{3} u + 1 - ln \frac{2}{3} u - 1))

代用を戻す

u = \frac{x}{6}

= 26 (\frac{1}{2} \cdot \frac{x}{6} - \frac{1}{4 6} (ln \frac{2}{3} \frac{x}{6} + 1 - ln \frac{2}{3} \frac{x}{6} - 1))

簡素化

26 (\frac{1}{2} \cdot \frac{x}{6} - \frac{1}{4 6} (ln \frac{2}{3} \frac{x}{6} + 1 - ln \frac{2}{3} \frac{x}{6} - 1)) : x - \frac{1}{2} ln \frac{x}{3} + 1 + \frac{1}{2} ln \frac{x}{3} - 1

= x - \frac{1}{2} ln \frac{x}{3} + 1 + \frac{1}{2} ln \frac{x}{3} - 1

定数を解答に追加する

= x - \frac{1}{2} ln \frac{x}{3} + 1 + \frac{1}{2} ln \frac{x}{3} - 1 + C

グラフ

人気の例

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