integral of sqrt(x)(4(x-1)^2+1/(x^3))

Lösung

\int x (4 (x - 1)^{2} + \frac{1}{x ^{3}}) d x

- \frac{2}{3 x ^{\frac{3}{2}}} + \frac{8}{7} x^{\frac{7}{2}} - \frac{16}{5} x^{\frac{5}{2}} + \frac{8}{3} x^{\frac{3}{2}} + C

Schritte zur Lösung

\int x (4 (x - 1)^{2} + \frac{1}{x ^{3}}) d x

Multipliziere aus

x (4 (x - 1)^{2} + \frac{1}{x ^{3}}) : \frac{1}{x ^{\frac{5}{2}}} + 4 x^{\frac{5}{2}} - 8 x^{\frac{3}{2}} + 4 x^{\frac{1}{2}}

= \int \frac{1}{x ^{\frac{5}{2}}} + 4 x^{\frac{5}{2}} - 8 x^{\frac{3}{2}} + 4 x^{\frac{1}{2}} 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 \frac{1}{x ^{\frac{5}{2}}} d x + \int 4 x^{\frac{5}{2}} d x - \int 8 x^{\frac{3}{2}} d x + \int 4 x^{\frac{1}{2}} d x

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

\int 4 x^{\frac{5}{2}} d x = \frac{8}{7} x^{\frac{7}{2}}

\int 8 x^{\frac{3}{2}} d x = \frac{16}{5} x^{\frac{5}{2}}

\int 4 x^{\frac{1}{2}} d x = \frac{8}{3} x^{\frac{3}{2}}

= - \frac{2}{3 x ^{\frac{3}{2}}} + \frac{8}{7} x^{\frac{7}{2}} - \frac{16}{5} x^{\frac{5}{2}} + \frac{8}{3} x^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{2}{3 x ^{\frac{3}{2}}} + \frac{8}{7} x^{\frac{7}{2}} - \frac{16}{5} x^{\frac{5}{2}} + \frac{8}{3} x^{\frac{3}{2}} + C

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