ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある微分積分 >

integral of sqrt(x^2-4x)

解

\int x^{2} - 4 x d x

解

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

解答ステップ

\int x^{2} - 4 x d x

平方完成する

x^{2} - 4 x : (x - 2)^{2} - 4

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

u置換積分法を適用する

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

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

integral from 1 to 3 of 3r^3ln(r)

\int_{1}^{3} 3 r^{3} ln (r) d r

integral of (7x-11)^4

\int (7 x - 11)^{4} d x

y^{''}-3y^'+2y=2t^3

y^{^{''}} - 3 y^{^{'}} + 2 y = 2 t^{3}

derivative of ln(x^4-6x^2)

\frac{d}{d x} (ln (x^{4} - 6 x^{2}))

integral of tan^3(21pix)

\int tan^{3} (21 π x) d x