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人気のある微分積分 >

integral of sqrt(y^2-x^2)

解

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

解

\frac{1}{2} y^{2} (arcsin (\frac{x}{y}) + \frac{1}{2} sin (2 arcsin (\frac{x}{y}))) + C

解答ステップ

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

三角関数による置換を適用する

= \int y^{2} cos^{2} (u) d u

定数を除く:

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

= y^{2} \cdot \int cos^{2} (u) d u

三角関数の公式を使用して書き換える

= y^{2} \cdot \int \frac{1 + cos ( 2 u )}{2} d u

定数を除く:

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

= y^{2} \frac{1}{2} \cdot \int 1 + cos (2 u) d u

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

= y^{2} \frac{1}{2} (\int 1 d u + \int cos (2 u) d u)

\int 1 d u = u

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

= y^{2} \frac{1}{2} (u + \frac{1}{2} sin (2 u))

代用を戻す

u = arcsin (\frac{1}{y} x)

= y^{2} \frac{1}{2} (arcsin (\frac{1}{y} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{y} x)))

簡素化

y^{2} \frac{1}{2} (arcsin (\frac{1}{y} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{y} x))) : \frac{1}{2} y^{2} (arcsin (\frac{x}{y}) + \frac{1}{2} sin (2 arcsin (\frac{x}{y})))

= \frac{1}{2} y^{2} (arcsin (\frac{x}{y}) + \frac{1}{2} sin (2 arcsin (\frac{x}{y})))

定数を解答に追加する

= \frac{1}{2} y^{2} (arcsin (\frac{x}{y}) + \frac{1}{2} sin (2 arcsin (\frac{x}{y}))) + C

グラフ

人気の例

limit as x approaches-2 of sqrt(25-5x)

x \to - 2 lim (25 - 5 x)

derivative f(x)=x^{4/3}-4x^{1/3}+1

d er i v a t i v e f (x) = x^{\frac{4}{3}} - 4 x^{\frac{1}{3}} + 1

derivative of x^2sqrt(5-x)

\frac{d}{d x} (x^{2} 5 - x)

derivative x^4sec(5/x)

d er i v a t i v e x^{4} sec (\frac{5}{x})

limit as z approaches 0 of 1/(f(z) 1/z)

z \to 0 lim (\frac{1}{f ( z ) \frac{1}{z}})