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人気のある微分積分 >

integral of (sqrt(x))/(x+2)

解

\int \frac{x}{x + 2} d x

解

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

解答ステップ

\int \frac{x}{x + 2} d x

u置換積分法を適用する

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

定数を除く:

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

= 2 \cdot \int \frac{u ^{2}}{u ^{2} + 2} d u

置換積分法を適用する

= 2 \cdot \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

= 22 \cdot \int \frac{v ^{2}}{v ^{2} + 1} d v

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

= 22 \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

= 22 (- \int \frac{1}{v ^{2} + 1} d v + \int 1 d v)

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

\int 1 d v = v

= 22 (- arctan (v) + v)

代用を戻す

= 22 (- arctan (\frac{x}{2}) + \frac{x}{2})

簡素化

22 (- arctan (\frac{x}{2}) + \frac{x}{2}) : 22 (- arctan (\frac{x}{2}) + \frac{x}{2})

= 22 (- arctan (\frac{x}{2}) + \frac{x}{2})

定数を解答に追加する

= 22 (- arctan (\frac{x}{2}) + \frac{x}{2}) + C

グラフ

人気の例

sum from n=1 to infinity of (n!)/(2n!)

n = 1 \sum \infty \frac{n !}{2 n !}

derivative of 2/(x^5)

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

derivative f(x)2z(9z^2-3z+1)^4

d er i v a t i v e f (x) 2 z (9 z^{2} - 3 z + 1)^{4}

integral from pi/6 to pi of sin(θ)

\int_{\frac{π}{6}}^{π} sin (θ) d θ

integral of 8/(cos(x)-1)

\int \frac{8}{cos ( x ) - 1} d x