integral of ((x^2+x+1))/((x+1)^2(x+2))

Lösung

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

- 2 ln ∣ x + 1 ∣ - \frac{1}{x + 1} + 3 ln ∣ x + 2 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{2} + x + 1}{( x + 1 ) ^{2} ( x + 2 )} : - \frac{2}{x + 1} + \frac{1}{( x + 1 ) ^{2}} + \frac{3}{x + 2}

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

\int \frac{2}{x + 1} d x = 2 ln ∣ x + 1 ∣

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

\int \frac{3}{x + 2} d x = 3 ln ∣ x + 2 ∣

= - 2 ln ∣ x + 1 ∣ - \frac{1}{x + 1} + 3 ln ∣ x + 2 ∣

Füge eine Konstante zur Lösung hinzu

= - 2 ln ∣ x + 1 ∣ - \frac{1}{x + 1} + 3 ln ∣ x + 2 ∣ + C

d er i v a t i v e y = a csc^{2} (k x)

\frac{d}{d x} (3 3 x)

ma c l a u r in \frac{1}{( 1 + x )}

((x + 1)^{2})^{^{'}}

\int \frac{1}{x ( 64 x ^{2} - 1 ) ^{\frac{1}{2}}} d x