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

Lösung

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

\frac{17}{9} ln ∣ x ∣ + \frac{5}{3 x} + \frac{1}{9} ln ∣ x + 3 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{2 x ^{2} + 4 x - 5}{x ^{2} ( x + 3 )} : \frac{17}{9 x} - \frac{5}{3 x ^{2}} + \frac{1}{9 ( x + 3 )}

= \int \frac{17}{9 x} - \frac{5}{3 x ^{2}} + \frac{1}{9 ( x + 3 )} 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{17}{9 x} d x - \int \frac{5}{3 x ^{2}} d x + \int \frac{1}{9 ( x + 3 )} d x

\int \frac{17}{9 x} d x = \frac{17}{9} ln ∣ x ∣

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

\int \frac{1}{9 ( x + 3 )} d x = \frac{1}{9} ln ∣ x + 3 ∣

= \frac{17}{9} ln ∣ x ∣ - (- \frac{5}{3 x}) + \frac{1}{9} ln ∣ x + 3 ∣

Vereinfache

= \frac{17}{9} ln ∣ x ∣ + \frac{5}{3 x} + \frac{1}{9} ln ∣ x + 3 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{17}{9} ln ∣ x ∣ + \frac{5}{3 x} + \frac{1}{9} ln ∣ x + 3 ∣ + C

\int_{1}^{3} x^{2} + 2 x - 2 d x

\int_{1}^{4} x^{\frac{3}{2}} ln (x) d x

\int_{0}^{1} \frac{1}{( x + 1 ) ^{2}} d x

f (t) = 3 t

\int_{0}^{1} \frac{x}{( 3 x ^{2} + 1 ) ^{4}} d x