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人気のある微分積分 >

integral of 4/((x^2)(x^2+4))

解

\int \frac{4}{( x ^{2} ) ( x ^{2} + 4 )} d x

解

- \frac{1}{x} - \frac{1}{2} arctan (\frac{x}{2}) + C

解答ステップ

\int \frac{4}{x ^{2} ( x ^{2} + 4 )} d x

定数を除く:

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

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

以下の部分分数を得る:

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

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

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

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

\int \frac{1}{4 x ^{2}} d x = - \frac{1}{4 x}

\int \frac{1}{4 ( x ^{2} + 4 )} d x = \frac{1}{8} arctan (\frac{x}{2})

= 4 (- \frac{1}{4 x} - \frac{1}{8} arctan (\frac{x}{2}))

簡素化

4 (- \frac{1}{4 x} - \frac{1}{8} arctan (\frac{x}{2})) : - \frac{1}{x} - \frac{1}{2} arctan (\frac{x}{2})

= - \frac{1}{x} - \frac{1}{2} arctan (\frac{x}{2})

定数を解答に追加する

= - \frac{1}{x} - \frac{1}{2} arctan (\frac{x}{2}) + C

グラフ

人気の例

面積 2x^2,8x^2,5-3x

a re a 2 x^{2}, 8 x^{2}, 5 - 3 x

(\partial)/(\partial x)(4x^2y+7xy^3)

\frac{\partial}{\partial x} (4 x^{2} y + 7 x y^{3})

derivative f(x)=(3x-2)(4x^3-x^2+1)

d er i v a t i v e f (x) = (3 x - 2) (4 x^{3} - x^{2} + 1)

limit as x approaches 2 of x^2-6x+3

x \to 2 lim (x^{2} - 6 x + 3)

derivative of 1/(1+e^{0.5x})

\frac{d}{d x} (\frac{1}{1 + e ^{0.5 x}})