integral of (5x^4+x^3-x^2+x+3)/(x^2)

Lösung

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

\frac{5 x ^{3}}{3} + \frac{x ^{2}}{2} - x + ln ∣ x ∣ - \frac{3}{x} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

\int 1 d x = x

\int \frac{1}{x} d x = ln ∣ x ∣

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

= \frac{5 x ^{3}}{3} + \frac{x ^{2}}{2} - x + ln ∣ x ∣ - \frac{3}{x}

Füge eine Konstante zur Lösung hinzu

= \frac{5 x ^{3}}{3} + \frac{x ^{2}}{2} - x + ln ∣ x ∣ - \frac{3}{x} + C

\int \frac{2 e ^{x}}{1 + e ^{2 x}} d x

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\int x e^{x} x d x

\int (3 - 4 x)^{11} d x

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