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(10-4x-4x^2)

解

\int 10 - 4 x - 4 x^{2} d x

解

\frac{11}{4} (arcsin (\frac{1}{11} (2 x + 1)) + \frac{1}{2} sin (2 arcsin (\frac{2 x + 1}{11}))) + C

解答ステップ

\int 10 - 4 x - 4 x^{2} d x

平方完成する

10 - 4 x - 4 x^{2} : - 4 (x + \frac{1}{2})^{2} + 11

= \int - 4 (x + \frac{1}{2})^{2} + 11 d x

u置換積分法を適用する

= \int - 4 u^{2} + 11 d u

三角関数による置換を適用する

= \int \frac{11}{2} cos^{2} (v) d v

定数を除く:

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

= \frac{11}{2} \cdot \int cos^{2} (v) d v

三角関数の公式を使用して書き換える

= \frac{11}{2} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

定数を除く:

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

= \frac{11}{2} \cdot \frac{1}{2} \cdot \int 1 + cos (2 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{11}{2} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= \frac{11}{2} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

代用を戻す

= \frac{11}{2} \cdot \frac{1}{2} (arcsin (\frac{2}{11} (x + \frac{1}{2})) + \frac{1}{2} sin (2 arcsin (\frac{2}{11} (x + \frac{1}{2}))))

簡素化

\frac{11}{2} \cdot \frac{1}{2} (arcsin (\frac{2}{11} (x + \frac{1}{2})) + \frac{1}{2} sin (2 arcsin (\frac{2}{11} (x + \frac{1}{2})))) : \frac{11}{4} (arcsin (\frac{1}{11} (2 x + 1)) + \frac{1}{2} sin (2 arcsin (\frac{2 x + 1}{11})))

= \frac{11}{4} (arcsin (\frac{1}{11} (2 x + 1)) + \frac{1}{2} sin (2 arcsin (\frac{2 x + 1}{11})))

定数を解答に追加する

= \frac{11}{4} (arcsin (\frac{1}{11} (2 x + 1)) + \frac{1}{2} sin (2 arcsin (\frac{2 x + 1}{11}))) + C

グラフ

人気の例

(\partial)/(\partial z)(2e^xcos(yz))

\frac{\partial}{\partial z} (2 e^{x} cos (yz))

limit as x approaches 0 of x^2-x

x \to 0 lim (x^{2} - x)

derivative of (6x^{3x})

\frac{d}{d x} ((6 x)^{3 x})

integral of sec(x)tan(x)

\int sec (x) tan (x) d x

y^{''}+5y^'=(4t-2)sin(2t)

y^{^{''}} + 5 y^{^{'}} = (4 t - 2) sin (2 t)