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人気のある微分積分 >

integral of sqrt(x(x-1))

解

\int x (x - 1) d x

解

\frac{1}{8} (4 x x (x - 1) - 2 x (x - 1) - ln 2 x - 1 + 2 x (x - 1)) + C

解答ステップ

\int x (x - 1) d x

拡張

x (x - 1) : x^{2} - x

= \int x^{2} - x d x

平方完成する

x^{2} - x : (x - \frac{1}{2})^{2} - \frac{1}{4}

= \int (x - \frac{1}{2})^{2} - \frac{1}{4} d x

u置換積分法を適用する

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

三角関数による置換を適用する

= \frac{1}{2} \cdot \int \frac{1}{2} 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

= \frac{1}{2} \cdot \frac{1}{2} \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

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

拡張

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

= \frac{1}{2} \cdot \frac{1}{2} \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

= \frac{1}{2} \cdot \frac{1}{2} (- \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) ∣

= \frac{1}{2} \cdot \frac{1}{2} (- 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

= \frac{1}{2} \cdot \frac{1}{2} (- ln tan (arcsec (2 (x - \frac{1}{2}))) + sec (arcsec (2 (x - \frac{1}{2}))) + \frac{1}{2} sec (arcsec (2 (x - \frac{1}{2}))) tan (arcsec (2 (x - \frac{1}{2}))) + \frac{1}{2} ln tan (arcsec (2 (x - \frac{1}{2}))) + sec (arcsec (2 (x - \frac{1}{2}))))

簡素化

\frac{1}{2} \cdot \frac{1}{2} (- ln tan (arcsec (2 (x - \frac{1}{2}))) + sec (arcsec (2 (x - \frac{1}{2}))) + \frac{1}{2} sec (arcsec (2 (x - \frac{1}{2}))) tan (arcsec (2 (x - \frac{1}{2}))) + \frac{1}{2} ln tan (arcsec (2 (x - \frac{1}{2}))) + sec (arcsec (2 (x - \frac{1}{2})))) : \frac{1}{8} (4 x x (x - 1) - 2 x (x - 1) - ln 2 x - 1 + 2 x (x - 1))

= \frac{1}{8} (4 x x (x - 1) - 2 x (x - 1) - ln 2 x - 1 + 2 x (x - 1))

定数を解答に追加する

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

グラフ

人気の例

integral of (x+1)/(x+4)

\int \frac{x + 1}{x + 4} d x

integral of 5x^2cos(3x)

\int 5 x^{2} cos (3 x) d x

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integral of sqrt((x-1)(x-2))

\int (x - 1) (x - 2) d x