integral of (x+6)/((x+3)(x+5))

Lösung

\int \frac{x + 6}{( x + 3 ) ( x + 5 )} d x

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

Schritte zur Lösung

\int \frac{x + 6}{( x + 3 ) ( x + 5 )} d x

Multipliziere aus

\frac{x + 6}{( x + 3 ) ( x + 5 )} : \frac{x}{( x + 3 ) ( x + 5 )} + \frac{6}{( x + 3 ) ( x + 5 )}

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{x}{( x + 3 ) ( x + 5 )} d x + \int \frac{6}{( x + 3 ) ( x + 5 )} d x

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

\int \frac{6}{( x + 3 ) ( x + 5 )} d x = 6 (\frac{1}{2} ln ∣ x + 3 ∣ - \frac{1}{2} ln ∣ x + 5 ∣)

= - \frac{3}{2} ln ∣ x + 3 ∣ + \frac{5}{2} ln ∣ x + 5 ∣ + 6 (\frac{1}{2} ln ∣ x + 3 ∣ - \frac{1}{2} ln ∣ x + 5 ∣)

Vereinfache

- \frac{3}{2} ln ∣ x + 3 ∣ + \frac{5}{2} ln ∣ x + 5 ∣ + 6 (\frac{1}{2} ln ∣ x + 3 ∣ - \frac{1}{2} ln ∣ x + 5 ∣) : \frac{3}{2} ln ∣ x + 3 ∣ - \frac{1}{2} ln ∣ x + 5 ∣

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

Füge eine Konstante zur Lösung hinzu

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

\int + \frac{8 x - 3}{12 x ^{2} - 7 x + 1} d x

\int \frac{4 cos ( x )}{( 3 + sin ( x ) ) ^{3}} d x

\int \frac{sin ^{3} ( x )}{2 x} d x

\int \frac{4 x}{1 - 2 x} d x

\int 4 t e^{7 t^{2}} d t