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Beliebt Rechnen >

integral of y^{-1/2}(y^4+2y^2-1)

Lösung

\int y^{- \frac{1}{2}} (y^{4} + 2 y^{2} - 1) d y

Lösung

\frac{2}{9} y^{\frac{9}{2}} + \frac{4}{5} y^{\frac{5}{2}} - 2 y^{\frac{1}{2}} + C

Schritte zur Lösung

\int y^{- \frac{1}{2}} (y^{4} + 2 y^{2} - 1) d y

Vereinfache

y^{- \frac{1}{2}} (y^{4} + 2 y^{2} - 1) : \frac{y ^{4} + 2 y ^{2} - 1}{y ^{\frac{1}{2}}}

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

Multipliziere aus

\frac{y ^{4} + 2 y ^{2} - 1}{y ^{\frac{1}{2}}} : y^{\frac{7}{2}} + 2 y^{\frac{3}{2}} - \frac{1}{y ^{\frac{1}{2}}}

= \int y^{\frac{7}{2}} + 2 y^{\frac{3}{2}} - \frac{1}{y ^{\frac{1}{2}}} d y

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int y^{\frac{7}{2}} d y + \int 2 y^{\frac{3}{2}} d y - \int \frac{1}{y ^{\frac{1}{2}}} d y

\int y^{\frac{7}{2}} d y = \frac{2}{9} y^{\frac{9}{2}}

\int 2 y^{\frac{3}{2}} d y = \frac{4}{5} y^{\frac{5}{2}}

\int \frac{1}{y ^{\frac{1}{2}}} d y = 2 y^{\frac{1}{2}}

= \frac{2}{9} y^{\frac{9}{2}} + \frac{4}{5} y^{\frac{5}{2}} - 2 y^{\frac{1}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{9} y^{\frac{9}{2}} + \frac{4}{5} y^{\frac{5}{2}} - 2 y^{\frac{1}{2}} + C

Graph

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