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

Lösung

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

ln x^{2} + 3 + 2 ln ∣ x ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

= 2 ln x^{2} + 3 + 2 ln ∣ x ∣ - ln x^{2} + 3

Vereinfache

= ln x^{2} + 3 + 2 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= ln x^{2} + 3 + 2 ln ∣ x ∣ + C

\int \frac{4 x ^{2} + 3 x + 19}{( x + 1 ) ( x ^{2} + 4 )} d x

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