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人気のある微分積分 >

integral of-x/(x+1)

解

\int - \frac{x}{x + 1} d x

解

- x - 1 + ln ∣ x + 1 ∣ + C

解答ステップ

\int - \frac{x}{x + 1} d x

定数を除く:

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

= - \int \frac{x}{x + 1} d x

u置換積分法を適用する

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

拡張

\frac{u - 1}{u} : 1 - \frac{1}{u}

= - \int 1 - \frac{1}{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

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

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= - (u - ln ∣ u ∣)

代用を戻す

u = x + 1

= - (x + 1 - ln ∣ x + 1 ∣)

簡素化

= - x - 1 + ln ∣ x + 1 ∣

定数を解答に追加する

= - x - 1 + ln ∣ x + 1 ∣ + C

グラフ

人気の例

tangent f(x)= 1/(x^6),(-2, 1/64)

t an g e n t f (x) = \frac{1}{x ^{6}}, (- 2, \frac{1}{64})

integral from 0 to 1 of 4/(x^2+3x+2)

\int_{0}^{1} \frac{4}{x ^{2} + 3 x + 2} d x

derivative of x^{2/5}(x-5)

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

y^{''}-6y^'=36t,y(0)=6,y^'(0)=0

y^{^{''}} - 6 y^{^{'}} = 36 t, y (0) = 6, y^{^{'}} (0) = 0

integral from 0 to 8 of x^3e^x

\int_{0}^{8} x^{3} e^{x} d x