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人気のある微分積分 >

integral of ((6-x))/(sqrt(4x^2-12x+7))

解

\int \frac{( 6 - x )}{4 x ^{2} - 12 x + 7} d x

解

\frac{1}{4} (- 4 x^{2} - 12 x + 7 + 9 ln 2 (x^{2} - 3 x + \frac{9}{4}) - 1 + 2 (x - \frac{3}{2})) + C

解答ステップ

\int \frac{6 - x}{4 x ^{2} - 12 x + 7} d x

平方完成する

4 x^{2} - 12 x + 7 : 4 (x - \frac{3}{2})^{2} - 2

= \int \frac{6 - x}{4 ( x - \frac{3}{2} ) ^{2} - 2} d x

u置換積分法を適用する

= \int \frac{- 2 u + 9}{2 4 u ^{2} - 2} d u

定数を除く:

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

= \frac{1}{2} \cdot \int \frac{- 2 u + 9}{4 u ^{2} - 2} d u

拡張

\frac{- 2 u + 9}{4 u ^{2} - 2} : - \frac{2 u}{4 u ^{2} - 2} + \frac{9}{4 u ^{2} - 2}

= \frac{1}{2} \cdot \int - \frac{2 u}{4 u ^{2} - 2} + \frac{9}{4 u ^{2} - 2} 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

= \frac{1}{2} (- \int \frac{2 u}{4 u ^{2} - 2} d u + \int \frac{9}{4 u ^{2} - 2} d u)

\int \frac{2 u}{4 u ^{2} - 2} d u = \frac{1}{2} 4 u^{2} - 2

\int \frac{9}{4 u ^{2} - 2} d u = \frac{9}{2} ln 2 u^{2} - 1 + 2 u

= \frac{1}{2} (- \frac{1}{2} 4 u^{2} - 2 + \frac{9}{2} ln 2 u^{2} - 1 + 2 u)

代用を戻す

u = x - \frac{3}{2}

= \frac{1}{2} - \frac{1}{2} 4 (x - \frac{3}{2})^{2} - 2 + \frac{9}{2} ln 2 (x - \frac{3}{2})^{2} - 1 + 2 (x - \frac{3}{2})

簡素化

\frac{1}{2} - \frac{1}{2} 4 (x - \frac{3}{2})^{2} - 2 + \frac{9}{2} ln 2 (x - \frac{3}{2})^{2} - 1 + 2 (x - \frac{3}{2}) : \frac{1}{4} (- 4 x^{2} - 12 x + 7 + 9 ln 2 (x^{2} - 3 x + \frac{9}{4}) - 1 + 2 (x - \frac{3}{2}))

= \frac{1}{4} (- 4 x^{2} - 12 x + 7 + 9 ln 2 (x^{2} - 3 x + \frac{9}{4}) - 1 + 2 (x - \frac{3}{2}))

定数を解答に追加する

= \frac{1}{4} (- 4 x^{2} - 12 x + 7 + 9 ln 2 (x^{2} - 3 x + \frac{9}{4}) - 1 + 2 (x - \frac{3}{2})) + C

グラフ

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