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Beliebt Rechnen >

integral of x/((1+x)^{3/4)}

Lösung

\int \frac{x}{( 1 + x ) ^{\frac{3}{4}}} d x

Lösung

\frac{4}{5} (1 + x)^{\frac{5}{4}} - 4 (1 + x)^{\frac{1}{4}} + C

Schritte zur Lösung

\int \frac{x}{( 1 + x ) ^{\frac{3}{4}}} d x

Wende U-Substitution an

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

Multipliziere aus

\frac{u - 1}{u ^{\frac{3}{4}}} : u^{\frac{1}{4}} - \frac{1}{u ^{\frac{3}{4}}}

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

\int u^{\frac{1}{4}} d u = \frac{4}{5} u^{\frac{5}{4}}

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

= \frac{4}{5} u^{\frac{5}{4}} - 4 u^{\frac{1}{4}}

Setze in

u = 1 + x

ein

= \frac{4}{5} (1 + x)^{\frac{5}{4}} - 4 (1 + x)^{\frac{1}{4}}

Füge eine Konstante zur Lösung hinzu

= \frac{4}{5} (1 + x)^{\frac{5}{4}} - 4 (1 + x)^{\frac{1}{4}} + C

Graph

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