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人気のある微分積分 >

integral of (-x+5)/(34(x^2+9))

解

\int \frac{- x + 5}{34 ( x ^{2} + 9 )} d x

解

\frac{1}{102} (- \frac{3}{2} ln \frac{x ^{2}}{9} + 1 + 5 arctan (\frac{x}{3})) + C

解答ステップ

\int \frac{- x + 5}{34 ( x ^{2} + 9 )} d x

定数を除く:

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

= \frac{1}{34} \cdot \int \frac{- x + 5}{x ^{2} + 9} d x

置換積分法を適用する

= \frac{1}{34} \cdot \int \frac{- 3 u + 5}{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}{34} \cdot \frac{1}{3} \cdot \int \frac{- 3 u + 5}{u ^{2} + 1} d u

拡張

\frac{- 3 u + 5}{u ^{2} + 1} : - \frac{3 u}{u ^{2} + 1} + \frac{5}{u ^{2} + 1}

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}{34} \cdot \frac{1}{3} (- \int \frac{3 u}{u ^{2} + 1} d u + \int \frac{5}{u ^{2} + 1} d u)

\int \frac{3 u}{u ^{2} + 1} d u = \frac{3}{2} ln u^{2} + 1

\int \frac{5}{u ^{2} + 1} d u = 5 arctan (u)

= \frac{1}{34} \cdot \frac{1}{3} (- \frac{3}{2} ln u^{2} + 1 + 5 arctan (u))

代用を戻す

u = \frac{x}{3}

= \frac{1}{34} \cdot \frac{1}{3} (- \frac{3}{2} ln (\frac{x}{3})^{2} + 1 + 5 arctan (\frac{x}{3}))

簡素化

\frac{1}{34} \cdot \frac{1}{3} (- \frac{3}{2} ln (\frac{x}{3})^{2} + 1 + 5 arctan (\frac{x}{3})) : \frac{1}{102} (- \frac{3}{2} ln \frac{x ^{2}}{9} + 1 + 5 arctan (\frac{x}{3}))

= \frac{1}{102} (- \frac{3}{2} ln \frac{x ^{2}}{9} + 1 + 5 arctan (\frac{x}{3}))

定数を解答に追加する

= \frac{1}{102} (- \frac{3}{2} ln \frac{x ^{2}}{9} + 1 + 5 arctan (\frac{x}{3})) + C

グラフ

人気の例

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