integral of ((7x^2+5x-6))/((x^2-4)(x+2))

Lösung

\int \frac{( 7 x ^{2} + 5 x - 6 )}{( x ^{2} - 4 ) ( x + 2 )} d x

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

Schritte zur Lösung

\int \frac{7 x ^{2} + 5 x - 6}{( x ^{2} - 4 ) ( x + 2 )} d x

Ermittle den Partialbruch von

\frac{7 x ^{2} + 5 x - 6}{( x ^{2} - 4 ) ( x + 2 )} : \frac{5}{x + 2} - \frac{3}{( x + 2 ) ^{2}} + \frac{2}{x - 2}

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

\int \frac{5}{x + 2} d x = 5 ln ∣ x + 2 ∣

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

\int \frac{2}{x - 2} d x = 2 ln ∣ x - 2 ∣

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d}{d x} (\frac{x}{( m + n x )})

\frac{\partial}{\partial x} (x^{2} + 2 x y + 2 y^{2} - 6 x + 10 y + 2)

d er i v a t i v e \frac{( x ^{2} + 1 )}{( x - 1 ) ^{2}}

\frac{\partial}{\partial x} (y x^{3} - x y^{2} + x y + 13 y)

\int \frac{x}{x ^{2} + y ^{2}} d y