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Beliebt Rechnen >

integral of ((5x+4))/(sqrt(5x-1))

Lösung

\int \frac{( 5 x + 4 )}{5 x - 1} d x

Lösung

\frac{2}{15} (5 x - 1)^{\frac{3}{2}} + 2 5 x - 1 + C

Schritte zur Lösung

\int \frac{5 x + 4}{5 x - 1} d x

Wende U-Substitution an

= \int \frac{2 ( u ^{2} + 5 )}{5} d u

Entferne die Konstante:

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

= \frac{2}{5} \cdot \int u^{2} + 5 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

= \frac{2}{5} (\int u^{2} d u + \int 5 d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int 5 d u = 5 u

= \frac{2}{5} (\frac{u ^{3}}{3} + 5 u)

Setze in

u = 5 x - 1

ein

= \frac{2}{5} (\frac{( 5 x - 1 ) ^{3}}{3} + 5 5 x - 1)

Vereinfache

\frac{2}{5} (\frac{( 5 x - 1 ) ^{3}}{3} + 5 5 x - 1) : \frac{2}{15} (5 x - 1)^{\frac{3}{2}} + 2 5 x - 1

= \frac{2}{15} (5 x - 1)^{\frac{3}{2}} + 2 5 x - 1

Füge eine Konstante zur Lösung hinzu

= \frac{2}{15} (5 x - 1)^{\frac{3}{2}} + 2 5 x - 1 + C

Graph

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