integral of (x^{1/4})/(sqrt(x)+1)

Lösung

\int \frac{x ^{\frac{1}{4}}}{x + 1} d x

4 (\frac{1}{3} x^{\frac{3}{4}} - 4 x + arctan (4 x)) + C

Schritte zur Lösung

\int \frac{x ^{\frac{1}{4}}}{x + 1} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

= 4 \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}

= 4 \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

= 4 (\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)

= 4 (\frac{u ^{3}}{3} - u + arctan (u))

Setze in

u = x^{\frac{1}{4}}

ein

= 4 \frac{( x ^{\frac{1}{4}} ) ^{3}}{3} - x^{\frac{1}{4}} + arctan (x^{\frac{1}{4}})

Vereinfache

4 \frac{( x ^{\frac{1}{4}} ) ^{3}}{3} - x^{\frac{1}{4}} + arctan (x^{\frac{1}{4}}) : 4 (\frac{1}{3} x^{\frac{3}{4}} - 4 x + arctan (4 x))

= 4 (\frac{1}{3} x^{\frac{3}{4}} - 4 x + arctan (4 x))

Füge eine Konstante zur Lösung hinzu

= 4 (\frac{1}{3} x^{\frac{3}{4}} - 4 x + arctan (4 x)) + C

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