integral of (4x^2+3x+6)/(x^2(x^2+3))

Lösung

\int \frac{4 x ^{2} + 3 x + 6}{x ^{2} ( x ^{2} + 3 )} d x

- \frac{1}{2} ln x^{2} + 3 + \frac{2}{3} arctan (\frac{x}{3}) + ln ∣ x ∣ - \frac{2}{x} + C

Schritte zur Lösung

\int \frac{4 x ^{2} + 3 x + 6}{x ^{2} ( x ^{2} + 3 )} d x

Multipliziere aus

\frac{4 x ^{2} + 3 x + 6}{x ^{2} ( x ^{2} + 3 )} : \frac{4 x ^{2}}{x ^{2} ( x ^{2} + 3 )} + \frac{3 x}{x ^{2} ( x ^{2} + 3 )} + \frac{6}{x ^{2} ( x ^{2} + 3 )}

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{4 x ^{2}}{x ^{2} ( x ^{2} + 3 )} d x + \int \frac{3 x}{x ^{2} ( x ^{2} + 3 )} d x + \int \frac{6}{x ^{2} ( x ^{2} + 3 )} d x

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

\int \frac{3 x}{x ^{2} ( x ^{2} + 3 )} d x = ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 3

\int \frac{6}{x ^{2} ( x ^{2} + 3 )} d x = 6 (- \frac{1}{3 x} - \frac{1}{3 3} arctan (\frac{x}{3}))

= \frac{4}{3} arctan (\frac{x}{3}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 3 + 6 (- \frac{1}{3 x} - \frac{1}{3 3} arctan (\frac{x}{3}))

Vereinfache

\frac{4}{3} arctan (\frac{x}{3}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 3 + 6 (- \frac{1}{3 x} - \frac{1}{3 3} arctan (\frac{x}{3})) : - \frac{1}{2} ln x^{2} + 3 + \frac{2}{3} arctan (\frac{x}{3}) + ln ∣ x ∣ - \frac{2}{x}

= - \frac{1}{2} ln x^{2} + 3 + \frac{2}{3} arctan (\frac{x}{3}) + ln ∣ x ∣ - \frac{2}{x}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln x^{2} + 3 + \frac{2}{3} arctan (\frac{x}{3}) + ln ∣ x ∣ - \frac{2}{x} + C