integral of 2/(x^2(x-5))

Lösung

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

2 (- \frac{1}{25} ln ∣ x ∣ + \frac{1}{5 x} + \frac{1}{25} ln ∣ x - 5 ∣) + C

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{1}{x ^{2} ( x - 5 )} d x

Ermittle den Partialbruch von

\frac{1}{x ^{2} ( x - 5 )} : - \frac{1}{25 x} - \frac{1}{5 x ^{2}} + \frac{1}{25 ( x - 5 )}

= 2 \cdot \int - \frac{1}{25 x} - \frac{1}{5 x ^{2}} + \frac{1}{25 ( 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

= 2 (- \int \frac{1}{25 x} d x - \int \frac{1}{5 x ^{2}} d x + \int \frac{1}{25 ( x - 5 )} d x)

\int \frac{1}{25 x} d x = \frac{1}{25} ln ∣ x ∣

\int \frac{1}{5 x ^{2}} d x = - \frac{1}{5 x}

\int \frac{1}{25 ( x - 5 )} d x = \frac{1}{25} ln ∣ x - 5 ∣

= 2 (- \frac{1}{25} ln ∣ x ∣ - (- \frac{1}{5 x}) + \frac{1}{25} ln ∣ x - 5 ∣)

Vereinfache

= 2 (- \frac{1}{25} ln ∣ x ∣ + \frac{1}{5 x} + \frac{1}{25} ln ∣ x - 5 ∣)

Füge eine Konstante zur Lösung hinzu

= 2 (- \frac{1}{25} ln ∣ x ∣ + \frac{1}{5 x} + \frac{1}{25} ln ∣ x - 5 ∣) + C

\int x^{3} cos (1 + x^{2}) d x

\int sin (x) cos (x) e^{- s i n (x)} d x

\int \frac{5}{4 x ^{2} + 25} d x

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

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