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人気のある微分積分 >

integral of 1/(a+bsqrt(x))

解

\int \frac{1}{a + b x} d x

解

\frac{2}{b ^{2}} (a + b x - a ln a + b x) + C

解答ステップ

\int \frac{1}{a + b x} d x

u置換積分法を適用する

= \int \frac{2 ( u - a )}{b ^{2} u} d u

定数を除く:

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

= \frac{2}{b ^{2}} \cdot \int \frac{u - a}{u} d u

拡張

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

= \frac{2}{b ^{2}} \cdot \int 1 - \frac{a}{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{2}{b ^{2}} (\int 1 d u - \int \frac{a}{u} d u)

\int 1 d u = u

\int \frac{a}{u} d u = a ln ∣ u ∣

= \frac{2}{b ^{2}} (u - a ln ∣ u ∣)

代用を戻す

u = a + b x

= \frac{2}{b ^{2}} (a + b x - a ln a + b x)

定数を解答に追加する

= \frac{2}{b ^{2}} (a + b x - a ln a + b x) + C

グラフ

人気の例

limit as x approaches 0 of (cot(x))/x

x \to 0 lim (\frac{cot ( x )}{x})

tangent y=3-2x^2,(-4,-29)

t an g e n t y = 3 - 2 x^{2}, (- 4, - 29)

derivative g(x)=3x^4

d er i v a t i v e g (x) = 3 x^{4}

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d er i v a t i v e f (x) = x^{3} + 5 x + 7

integral from 1 to e of ln(13x)

\int_{1}^{e} ln (13 x) d x