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人気のある微分積分 >

integral of x/(x+\sqrt[3]{x)}

解

\int \frac{x}{x + 3 x} d x

解

x - 3 3 x + 3 arctan (3 x) + C

解答ステップ

\int \frac{x}{x + 3 x} d x

u置換積分法を適用する

= \int \frac{3 u ^{4}}{u ^{2} + 1} d u

定数を除く:

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

= 3 \cdot \int \frac{u ^{4}}{u ^{2} + 1} d u

長除法

\frac{u ^{4}}{u ^{2} + 1} : u^{2} - 1 + \frac{1}{u ^{2} + 1}

= 3 \cdot \int u^{2} - 1 + \frac{1}{u ^{2} + 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

= 3 (\int u^{2} d u - \int 1 d u + \int \frac{1}{u ^{2} + 1} d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int 1 d u = u

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

= 3 (\frac{u ^{3}}{3} - u + arctan (u))

代用を戻す

u = 3 x

= 3 (\frac{( 3 x ) ^{3}}{3} - 3 x + arctan (3 x))

簡素化

3 (\frac{( 3 x ) ^{3}}{3} - 3 x + arctan (3 x)) : x - 3 3 x + 3 arctan (3 x)

= x - 3 3 x + 3 arctan (3 x)

定数を解答に追加する

= x - 3 3 x + 3 arctan (3 x) + C

グラフ

人気の例

derivative of (1/(x^2-9/(x^4))(x+5x^3))

\frac{d}{d x} ((\frac{1}{x ^{2}} - \frac{9}{x ^{4}}) (x + 5 x^{3}))

limit as x approaches 0 of 2e^{3x}

x \to 0 lim (2 e^{3 x})

tangent f(x)=(12x)/(x^2-4),(4,4)

t an g e n t f (x) = \frac{12 x}{x ^{2} - 4}, (4, 4)

integral of 2/(x^2sqrt(1+x^2))

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

derivative (8x^2+9)/(x^2+5)

d er i v a t i v e \frac{8 x ^{2} + 9}{x ^{2} + 5}