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Beliebt Rechnen >

integral of ((y+3))/((3-y)^{2/3)}

Lösung

\int \frac{( y + 3 )}{( 3 - y ) ^{\frac{2}{3}}} d y

Lösung

\frac{3}{4} (3 - y)^{\frac{4}{3}} - 18 (3 - y)^{\frac{1}{3}} + C

Schritte zur Lösung

\int \frac{y + 3}{( 3 - y ) ^{\frac{2}{3}}} d y

Wende U-Substitution an

= \int - \frac{- u + 6}{u ^{\frac{2}{3}}} d u

Entferne die Konstante:

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

= - \int \frac{- u + 6}{u ^{\frac{2}{3}}} d u

Multipliziere aus

\frac{- u + 6}{u ^{\frac{2}{3}}} : - u^{\frac{1}{3}} + \frac{6}{u ^{\frac{2}{3}}}

= - \int - u^{\frac{1}{3}} + \frac{6}{u ^{\frac{2}{3}}} 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}{3}} d u + \int \frac{6}{u ^{\frac{2}{3}}} d u)

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

\int \frac{6}{u ^{\frac{2}{3}}} d u = 18 u^{\frac{1}{3}}

= - (- \frac{3}{4} u^{\frac{4}{3}} + 18 u^{\frac{1}{3}})

Setze in

u = 3 - y

ein

= - (- \frac{3}{4} (3 - y)^{\frac{4}{3}} + 18 (3 - y)^{\frac{1}{3}})

Vereinfache

= \frac{3}{4} (3 - y)^{\frac{4}{3}} - 18 (3 - y)^{\frac{1}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{4} (3 - y)^{\frac{4}{3}} - 18 (3 - y)^{\frac{1}{3}} + C

Graph

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