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人気のある微分積分 >

integral of sqrt(1-100x^2)

解

\int 1 - 100 x^{2} d x

解

\frac{1}{20} (arcsin (10 x) + \frac{1}{2} sin (2 arcsin (10 x))) + C

解答ステップ

\int 1 - 100 x^{2} d x

三角関数による置換を適用する

= \int \frac{1}{10} cos^{2} (u) d u

定数を除く:

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

= \frac{1}{10} \cdot \int cos^{2} (u) d u

三角関数の公式を使用して書き換える

= \frac{1}{10} \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

= \frac{1}{10} \cdot \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

= \frac{1}{10} \cdot \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)

= \frac{1}{10} \cdot \frac{1}{2} (u + \frac{1}{2} sin (2 u))

代用を戻す

u = arcsin (10 x)

= \frac{1}{10} \cdot \frac{1}{2} (arcsin (10 x) + \frac{1}{2} sin (2 arcsin (10 x)))

簡素化

\frac{1}{10} \cdot \frac{1}{2} (arcsin (10 x) + \frac{1}{2} sin (2 arcsin (10 x))) : \frac{1}{20} (arcsin (10 x) + \frac{1}{2} sin (2 arcsin (10 x)))

= \frac{1}{20} (arcsin (10 x) + \frac{1}{2} sin (2 arcsin (10 x)))

定数を解答に追加する

= \frac{1}{20} (arcsin (10 x) + \frac{1}{2} sin (2 arcsin (10 x))) + C

グラフ

人気の例

integral from 0 to pi/2 of 10sin^2(3x)

\int_{0}^{\frac{π}{2}} 10 sin^{2} (3 x) d x

limit as x approaches 1+of xln(x)

x \to 1 + lim (x ln (x))

derivative of \sqrt[3]{3x^2}

\frac{d}{d x} (3 3 x^{2})

limit as x approaches 8 of 1+\sqrt[3]{x}

x \to 8 lim (1 + 3 x)

x^2y^'-x(x+2)y=e^x

x^{2} y^{^{'}} - x (x + 2) y = e^{x}