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人気のある微分積分 >

integral of sqrt(x)sqrt(49-x)

解

\int x 49 - x d x

解

\frac{2401}{16} (2 arcsin (\frac{2}{49} x - 1) + sin (2 arcsin (\frac{2}{49} x - 1))) + C

解答ステップ

\int x 49 - x d x

指数の規則を適用する:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

x 49 - x = x (49 - x)

= \int x (49 - x) d x

拡張

x (49 - x) : 49 x - x^{2}

= \int 49 x - x^{2} d x

平方完成する

49 x - x^{2} : - (x - \frac{49}{2})^{2} + \frac{2401}{4}

= \int - (x - \frac{49}{2})^{2} + \frac{2401}{4} d x

u置換積分法を適用する

= \int \frac{- 4 u ^{2} + 2401}{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} + 2401 d u

三角関数による置換を適用する

= \frac{1}{2} \cdot \int \frac{2401}{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{1}{2} \cdot \frac{2401}{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{1}{2} \cdot \frac{2401}{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{1}{2} \cdot \frac{2401}{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{1}{2} \cdot \frac{2401}{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{1}{2} \cdot \frac{2401}{2} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

代用を戻す

= \frac{1}{2} \cdot \frac{2401}{2} \cdot \frac{1}{2} (arcsin (\frac{2}{49} (x - \frac{49}{2})) + \frac{1}{2} sin (2 arcsin (\frac{2}{49} (x - \frac{49}{2}))))

簡素化

\frac{1}{2} \cdot \frac{2401}{2} \cdot \frac{1}{2} (arcsin (\frac{2}{49} (x - \frac{49}{2})) + \frac{1}{2} sin (2 arcsin (\frac{2}{49} (x - \frac{49}{2})))) : \frac{2401}{16} (2 arcsin (\frac{2}{49} x - 1) + sin (2 arcsin (\frac{2}{49} x - 1)))

= \frac{2401}{16} (2 arcsin (\frac{2}{49} x - 1) + sin (2 arcsin (\frac{2}{49} x - 1)))

定数を解答に追加する

= \frac{2401}{16} (2 arcsin (\frac{2}{49} x - 1) + sin (2 arcsin (\frac{2}{49} x - 1))) + C

グラフ

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