integral of x/(\sqrt[4]{x+3)}

Lösung

\int \frac{x}{4 x + 3} d x

\frac{4}{7} (x + 3)^{\frac{7}{4}} - 4 (x + 3)^{\frac{3}{4}} + C

Schritte zur Lösung

\int \frac{x}{4 x + 3} d x

Wende U-Substitution an

= \int \frac{u - 3}{4 u} d u

Multipliziere aus

\frac{u - 3}{4 u} : u^{\frac{3}{4}} - \frac{3}{4 u}

= \int u^{\frac{3}{4}} - \frac{3}{4 u} 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

= \int u^{\frac{3}{4}} d u - \int \frac{3}{4 u} d u

\int u^{\frac{3}{4}} d u = \frac{4}{7} u^{\frac{7}{4}}

\int \frac{3}{4 u} d u = 4 u^{\frac{3}{4}}

= \frac{4}{7} u^{\frac{7}{4}} - 4 u^{\frac{3}{4}}

Setze in

u = x + 3

ein

= \frac{4}{7} (x + 3)^{\frac{7}{4}} - 4 (x + 3)^{\frac{3}{4}}

Füge eine Konstante zur Lösung hinzu

= \frac{4}{7} (x + 3)^{\frac{7}{4}} - 4 (x + 3)^{\frac{3}{4}} + C

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