integral of (12x^2+4x-16)/(6x^2-13x-5)

Lösung

\int \frac{12 x ^{2} + 4 x - 16}{6 x ^{2} - 13 x - 5} d x

2 x + \frac{16}{17} ln ∣ 3 x + 1 ∣ + \frac{69}{17} ln ∣ 2 x - 5 ∣ + C

Schritte zur Lösung

\int \frac{12 x ^{2} + 4 x - 16}{6 x ^{2} - 13 x - 5} d x

Ermittle den Partialbruch von

\frac{12 x ^{2} + 4 x - 16}{6 x ^{2} - 13 x - 5} : 2 + \frac{48}{17 ( 3 x + 1 )} + \frac{138}{17 ( 2 x - 5 )}

= \int 2 + \frac{48}{17 ( 3 x + 1 )} + \frac{138}{17 ( 2 x - 5 )} 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 2 d x + \int \frac{48}{17 ( 3 x + 1 )} d x + \int \frac{138}{17 ( 2 x - 5 )} d x

\int 2 d x = 2 x

\int \frac{48}{17 ( 3 x + 1 )} d x = \frac{16}{17} ln ∣ 3 x + 1 ∣

\int \frac{138}{17 ( 2 x - 5 )} d x = \frac{69}{17} ln ∣ 2 x - 5 ∣

= 2 x + \frac{16}{17} ln ∣ 3 x + 1 ∣ + \frac{69}{17} ln ∣ 2 x - 5 ∣

Füge eine Konstante zur Lösung hinzu

= 2 x + \frac{16}{17} ln ∣ 3 x + 1 ∣ + \frac{69}{17} ln ∣ 2 x - 5 ∣ + C

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

\int - 3 x^{2} + 2 x d x

\int \frac{x ^{2} + 1}{x ^{6}} d x

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

\int (5 y) d y