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人気のある微分積分 >

integral of (x^2-6x+9)/(sqrt(x^2-6x+13))

解

\int \frac{x ^{2} - 6 x + 9}{x ^{2} - 6 x + 13} d x

解

4 (- ln (\frac{1}{2} x - 3 + x^{2} - 6 x + 13) + \frac{1}{8} (x - 3) x^{2} - 6 x + 13 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 13}{2})) + C

解答ステップ

\int \frac{x ^{2} - 6 x + 9}{x ^{2} - 6 x + 13} d x

平方完成する

x^{2} - 6 x + 9 : (x - 3)^{2}

= \int \frac{( x - 3 ) ^{2}}{x ^{2} - 6 x + 13} d x

u置換積分法を適用する

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

三角関数による置換を適用する

= \int 4 tan^{2} (v) sec (v) d v

定数を除く:

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

= 4 \cdot \int tan^{2} (v) sec (v) d v

三角関数の公式を使用して書き換える

= 4 \cdot \int (- 1 + sec^{2} (v)) sec (v) d v

拡張

(- 1 + sec^{2} (v)) sec (v) : - sec (v) + sec^{3} (v)

= 4 \cdot \int - sec (v) + sec^{3} (v) d v

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

= 4 (- \int sec (v) d v + \int sec^{3} (v) d v)

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

\int sec^{3} (v) d v = \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣

= 4 (- ln ∣ tan (v) + sec (v) ∣ + \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

代用を戻す

= 4 (- ln tan (arctan (\frac{1}{2} (x - 3))) + sec (arctan (\frac{1}{2} (x - 3))) + \frac{1}{2} sec (arctan (\frac{1}{2} (x - 3))) tan (arctan (\frac{1}{2} (x - 3))) + \frac{1}{2} ln tan (arctan (\frac{1}{2} (x - 3))) + sec (arctan (\frac{1}{2} (x - 3))))

簡素化

4 (- ln tan (arctan (\frac{1}{2} (x - 3))) + sec (arctan (\frac{1}{2} (x - 3))) + \frac{1}{2} sec (arctan (\frac{1}{2} (x - 3))) tan (arctan (\frac{1}{2} (x - 3))) + \frac{1}{2} ln tan (arctan (\frac{1}{2} (x - 3))) + sec (arctan (\frac{1}{2} (x - 3)))) : 4 (- ln (\frac{1}{2} x - 3 + x^{2} - 6 x + 13) + \frac{1}{8} (x - 3) x^{2} - 6 x + 13 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 13}{2}))

= 4 (- ln (\frac{1}{2} x - 3 + x^{2} - 6 x + 13) + \frac{1}{8} (x - 3) x^{2} - 6 x + 13 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 13}{2}))

定数を解答に追加する

= 4 (- ln (\frac{1}{2} x - 3 + x^{2} - 6 x + 13) + \frac{1}{8} (x - 3) x^{2} - 6 x + 13 + \frac{1}{2} ln (\frac{x - 3 + x ^{2} - 6 x + 13}{2})) + C

グラフ

人気の例

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\int \frac{90 x ^{2}}{x ^{4} - 53 x ^{2} + 196} d x

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