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(3x-2))

解

\int x (3 x - 2) d x

解

\frac{1}{6 3} ((3 x - 1) 9 x^{2} - 6 x - ln 3 x - 1 + 9 x^{2} - 6 x) + C

解答ステップ

\int x (3 x - 2) d x

平方完成する

x (3 x - 2) : 3 (x - \frac{1}{3})^{2} - \frac{1}{3}

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

u置換積分法を適用する

= \int \frac{9 u ^{2} - 1}{3} d u

\frac{9 u ^{2} - 1}{3} = \frac{1}{3} 9 u^{2} - 1

= \int \frac{1}{3} 9 u^{2} - 1 d u

定数を除く:

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

= \frac{1}{3} \cdot \int 9 u^{2} - 1 d u

三角関数による置換を適用する

= \frac{1}{3} \cdot \int \frac{1}{3} 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}{3} \frac{1}{3} \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}{3} \frac{1}{3} \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}{3} \frac{1}{3} \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}{3} \frac{1}{3} (- \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}{3} \frac{1}{3} (- 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}{3} \frac{1}{3} (- ln tan (arcsec (3 (x - \frac{1}{3}))) + sec (arcsec (3 (x - \frac{1}{3}))) + \frac{1}{2} sec (arcsec (3 (x - \frac{1}{3}))) tan (arcsec (3 (x - \frac{1}{3}))) + \frac{1}{2} ln tan (arcsec (3 (x - \frac{1}{3}))) + sec (arcsec (3 (x - \frac{1}{3}))))

簡素化

\frac{1}{3} \frac{1}{3} (- ln tan (arcsec (3 (x - \frac{1}{3}))) + sec (arcsec (3 (x - \frac{1}{3}))) + \frac{1}{2} sec (arcsec (3 (x - \frac{1}{3}))) tan (arcsec (3 (x - \frac{1}{3}))) + \frac{1}{2} ln tan (arcsec (3 (x - \frac{1}{3}))) + sec (arcsec (3 (x - \frac{1}{3})))) : \frac{1}{6 3} ((3 x - 1) 9 x^{2} - 6 x - ln 3 x - 1 + 9 x^{2} - 6 x)

= \frac{1}{6 3} ((3 x - 1) 9 x^{2} - 6 x - ln 3 x - 1 + 9 x^{2} - 6 x)

定数を解答に追加する

= \frac{1}{6 3} ((3 x - 1) 9 x^{2} - 6 x - ln 3 x - 1 + 9 x^{2} - 6 x) + C

グラフ

人気の例

(\partial)/(\partial y)(2x^2)

\frac{\partial}{\partial y} (2 x^{2})

integral of (2x^3+2x^2+2x+1)/(x^2+x)

\int \frac{2 x ^{3} + 2 x ^{2} + 2 x + 1}{x ^{2} + x} d x

tangent y=4x^3-2x

t an g e n t y = 4 x^{3} - 2 x

limit as x approaches 5 of x(5)

x \to 5 lim (x (5))

derivative of (-2/(sqrt(x)))

\frac{d}{d x} (\frac{- 2}{x})