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人気のある微分積分 >

integral of sqrt((4-x^2)/(x^2))

解

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

解

- ln \frac{1}{2} 4 - x^{2} + 1 + ln \frac{1}{2} 4 - x^{2} - 1 + 4 - x^{2} + C

解答ステップ

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

簡素化

\frac{4 - x ^{2}}{x ^{2}} : \frac{4 - x ^{2}}{x ^{2}}

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

x^{2} = (x), （ x \geq 0

と仮定

）

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

u置換積分法を適用する

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

定数を除く:

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

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

置換積分法を適用する

= - \int \frac{2 v ^{2}}{- v ^{2} + 1} d v

定数を除く:

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

= - 2 \cdot \int \frac{v ^{2}}{- v ^{2} + 1} d v

\frac{v ^{2}}{- v ^{2} + 1} = \frac{1}{- v ^{2} + 1} - 1

= - 2 \cdot \int \frac{1}{- v ^{2} + 1} - 1 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

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

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

\int 1 d v = v

= - 2 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2} - v)

代用を戻す

= - 2 \frac{ln \frac{4 - x ^{2}}{2} + 1}{2} - \frac{ln \frac{4 - x ^{2}}{2} - 1}{2} - \frac{4 - x ^{2}}{2}

簡素化

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

= - ln \frac{1}{2} 4 - x^{2} + 1 + ln \frac{1}{2} 4 - x^{2} - 1 + 4 - x^{2}

定数を解答に追加する

= - ln \frac{1}{2} 4 - x^{2} + 1 + ln \frac{1}{2} 4 - x^{2} - 1 + 4 - x^{2} + C

グラフ

人気の例

integral from 0 to 11 of xsqrt(11-x)

\int_{0}^{11} x 11 - x d x

limit as x approaches 3 of 5x^3

x \to 3 lim (5 x^{3})

limit as x approaches-1 of e^{x+1}

x \to - 1 lim (e^{x + 1})

derivative 10x(x^2+6)^4

d er i v a t i v e 10 x (x^{2} + 6)^{4}

limit as x approaches 2-of sqrt(x(x-1))

x \to 2 - lim (x (x - 1))