integral of ((2x-3)^3)/(sqrt(x))

Lösung

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

\frac{16}{7} x^{\frac{7}{2}} - \frac{72}{5} x^{\frac{5}{2}} + 36 x^{\frac{3}{2}} - 54 x + C

Schritte zur Lösung

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

Multipliziere aus

\frac{( 2 x - 3 ) ^{3}}{x} : 8 x^{\frac{5}{2}} - 36 x^{\frac{3}{2}} + 54 x - \frac{27}{x}

= \int 8 x^{\frac{5}{2}} - 36 x^{\frac{3}{2}} + 54 x - \frac{27}{x} d x

Wende die Summenregel an:

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

= \int 8 x^{\frac{5}{2}} d x - \int 36 x^{\frac{3}{2}} d x + \int 54 x d x - \int \frac{27}{x} d x

\int 8 x^{\frac{5}{2}} d x = \frac{16}{7} x^{\frac{7}{2}}

\int 36 x^{\frac{3}{2}} d x = \frac{72}{5} x^{\frac{5}{2}}

\int 54 x d x = 36 x^{\frac{3}{2}}

\int \frac{27}{x} d x = 54 x

= \frac{16}{7} x^{\frac{7}{2}} - \frac{72}{5} x^{\frac{5}{2}} + 36 x^{\frac{3}{2}} - 54 x

Füge eine Konstante zur Lösung hinzu

= \frac{16}{7} x^{\frac{7}{2}} - \frac{72}{5} x^{\frac{5}{2}} + 36 x^{\frac{3}{2}} - 54 x + C

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