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

Lösung

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

\frac{2}{7} x^{\frac{7}{2}} - 3 ln ∣ x ∣ - \frac{4}{x ^{\frac{1}{2}}} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

= \frac{2}{7} x^{\frac{7}{2}} - 3 ln ∣ x ∣ - \frac{4}{x ^{\frac{1}{2}}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{7} x^{\frac{7}{2}} - 3 ln ∣ x ∣ - \frac{4}{x ^{\frac{1}{2}}} + C

d er i v a t i v e f (x) = \frac{x ^{3}}{5 - x ^{2}}

\int_{0}^{2 π} sin (x) cos^{2} (x) d x

\int sec (y) (tan (y) - sec (y)) d y

\frac{d}{d x} (e^{(x + 1)^{2}})

x \to π lim (ln (cos (x) - 5))