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人気のある微分積分 >

integral of x/(sqrt(x^2-6x+8))

解

\int \frac{x}{x ^{2} - 6 x + 8} d x

解

x^{2} - 6 x + 8 + 3 ln x^{2} - 6 x + 8 + x - 3 + C

解答ステップ

\int \frac{x}{x ^{2} - 6 x + 8} d x

平方完成する

x^{2} - 6 x + 8 : (x - 3)^{2} - 1

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

u置換積分法を適用する

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

拡張

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

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

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

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

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

= u^{2} - 1 + 3 ln u^{2} - 1 + u

代用を戻す

u = x - 3

= (x - 3)^{2} - 1 + 3 ln (x - 3)^{2} - 1 + x - 3

簡素化

(x - 3)^{2} - 1 + 3 ln (x - 3)^{2} - 1 + x - 3 : x^{2} - 6 x + 8 + 3 ln x^{2} - 6 x + 8 + x - 3

= x^{2} - 6 x + 8 + 3 ln x^{2} - 6 x + 8 + x - 3

定数を解答に追加する

= x^{2} - 6 x + 8 + 3 ln x^{2} - 6 x + 8 + x - 3 + C

グラフ

人気の例

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