integral of ((x+1)^3)/(sqrt(x))

Lösung

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

\frac{2}{7} x^{\frac{7}{2}} + \frac{6}{5} x^{\frac{5}{2}} + 2 x^{\frac{3}{2}} + 2 x + C

Schritte zur Lösung

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

Multipliziere aus

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

= \int x^{\frac{5}{2}} + 3 x^{\frac{3}{2}} + 3 x + \frac{1}{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 x^{\frac{5}{2}} d x + \int 3 x^{\frac{3}{2}} d x + \int 3 x d x + \int \frac{1}{x} d x

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

\int 3 x^{\frac{3}{2}} d x = \frac{6}{5} x^{\frac{5}{2}}

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

\int \frac{1}{x} d x = 2 x

= \frac{2}{7} x^{\frac{7}{2}} + \frac{6}{5} x^{\frac{5}{2}} + 2 x^{\frac{3}{2}} + 2 x

Füge eine Konstante zur Lösung hinzu

= \frac{2}{7} x^{\frac{7}{2}} + \frac{6}{5} x^{\frac{5}{2}} + 2 x^{\frac{3}{2}} + 2 x + C