integral of x(\sqrt[3]{x}+sqrt(x)+1)

Lösung

\int x (3 x + x + 1) d x

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

Schritte zur Lösung

\int x (3 x + x + 1) d x

Multipliziere aus

x (3 x + x + 1) : x^{\frac{4}{3}} + x^{\frac{3}{2}} + x

= \int x^{\frac{4}{3}} + x^{\frac{3}{2}} + 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{4}{3}} d x + \int x^{\frac{3}{2}} d x + \int x d x

\int x^{\frac{4}{3}} d x = \frac{3}{7} x^{\frac{7}{3}}

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

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

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e f (x) = sin (x) ln (x^{2} + 1)

x \to π lim (9 cot (x))

d er i v a t i v e y = ln (x^{6})

x \to 0 lim ((e^{x} - x)^{\frac{1}{x}})

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