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人気のある微分積分 >

integral of (sqrt(1-4x^2))/x

解

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

解

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

解答ステップ

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

u置換積分法を適用する

= \int \frac{u ^{2}}{u ^{2} - 1} d u

因数

u^{2} - 1 : - (- u^{2} + 1)

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

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

= - \int \frac{1}{- u ^{2} + 1} - 1 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

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

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

\int 1 d u = u

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

代用を戻す

u = 1 - 4 x^{2}

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

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

derivative of (2x^2+2x+6/(sqrt(x)))

\frac{d}{d x} (\frac{2 x ^{2} + 2 x + 6}{x})

derivative 2/(3t^2)

d er i v a t i v e \frac{2}{3 t ^{2}}

integral of 1/2 x^4

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

derivative (440t)/((t^2+6)^3)

d er i v a t i v e \frac{440 t}{( t ^{2} + 6 ) ^{3}}

tangent (7x)/(x+4)(3.3)

t an g e n t \frac{7 x}{x + 4} (3.3)