integral of (50)/((x-1)(x^2+49))

Lösung

\int \frac{50}{( x - 1 ) ( x ^{2} + 49 )} d x

ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7}) + C

Schritte zur Lösung

\int \frac{50}{( x - 1 ) ( x ^{2} + 49 )} d x

Entferne die Konstante:

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

= 50 \cdot \int \frac{1}{( x - 1 ) ( x ^{2} + 49 )} d x

Ermittle den Partialbruch von

\frac{1}{( x - 1 ) ( x ^{2} + 49 )} : \frac{1}{50 ( x - 1 )} + \frac{- x - 1}{50 ( x ^{2} + 49 )}

= 50 \cdot \int \frac{1}{50 ( x - 1 )} + \frac{- x - 1}{50 ( x ^{2} + 49 )} 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

= 50 (\int \frac{1}{50 ( x - 1 )} d x + \int \frac{- x - 1}{50 ( x ^{2} + 49 )} d x)

\int \frac{1}{50 ( x - 1 )} d x = \frac{1}{50} ln ∣ x - 1 ∣

\int \frac{- x - 1}{50 ( x ^{2} + 49 )} d x = \frac{1}{50} (- \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7}))

= 50 (\frac{1}{50} ln ∣ x - 1 ∣ + \frac{1}{50} (- \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7})))

Vereinfache

50 (\frac{1}{50} ln ∣ x - 1 ∣ + \frac{1}{50} (- \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7}))) : ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7})

= ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7})

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 49 - \frac{1}{7} arctan (\frac{x}{7}) + C

d er i v a t i v e ln (e^{x^{2}})

\int \frac{x ^{2}}{x ^{2} - 4 x + 13} d x

\int \frac{e ^{4 x}}{2} d x

x \to 0 lim (x \cdot ln (x))

x \to 3 lim (4^{f (x)})