integral of 1/((x-3)(x^2+1))

Lösung

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

\frac{1}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (- \frac{1}{2} ln x^{2} + 1 - 3 arctan (x)) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{( x - 3 ) ( x ^{2} + 1 )} : \frac{1}{10 ( x - 3 )} + \frac{- x - 3}{10 ( x ^{2} + 1 )}

= \int \frac{1}{10 ( x - 3 )} + \frac{- x - 3}{10 ( x ^{2} + 1 )} 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{1}{10 ( x - 3 )} d x + \int \frac{- x - 3}{10 ( x ^{2} + 1 )} d x

\int \frac{1}{10 ( x - 3 )} d x = \frac{1}{10} ln ∣ x - 3 ∣

\int \frac{- x - 3}{10 ( x ^{2} + 1 )} d x = \frac{1}{10} (- \frac{1}{2} ln x^{2} + 1 - 3 arctan (x))

= \frac{1}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (- \frac{1}{2} ln x^{2} + 1 - 3 arctan (x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{10} ln ∣ x - 3 ∣ + \frac{1}{10} (- \frac{1}{2} ln x^{2} + 1 - 3 arctan (x)) + C

x \to 2 + lim (\frac{x + 7}{x ^{2} - 4})

x \to 0 lim (\frac{1 - cos ( 9 x )}{x})

d er i v a t i v e f (x) e^{6} + ln (4)

\int_{0}^{\frac{π}{2}} 17 sin^{2} (2 x) d x

\frac{\partial}{\partial y} (ln (x - 4 y))