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

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

\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

Lange Division

\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

Wende die Summenregel an:

\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))

Setze in

u = 3 x

ein

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

Vereinfache

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)

Füge eine Konstante zur Lösung hinzu

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

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

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

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

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

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