integral of (26)/((x-1)(x^2+25))

Lösung

\int \frac{26}{( x - 1 ) ( x ^{2} + 25 )} d x

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

Schritte zur Lösung

\int \frac{26}{( x - 1 ) ( x ^{2} + 25 )} d x

Entferne die Konstante:

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

= 26 \cdot \int \frac{1}{( x - 1 ) ( x ^{2} + 25 )} d x

Ermittle den Partialbruch von

\frac{1}{( x - 1 ) ( x ^{2} + 25 )} : \frac{1}{26 ( x - 1 )} + \frac{- x - 1}{26 ( x ^{2} + 25 )}

= 26 \cdot \int \frac{1}{26 ( x - 1 )} + \frac{- x - 1}{26 ( x ^{2} + 25 )} 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

= 26 (\int \frac{1}{26 ( x - 1 )} d x + \int \frac{- x - 1}{26 ( x ^{2} + 25 )} d x)

\int \frac{1}{26 ( x - 1 )} d x = \frac{1}{26} ln ∣ x - 1 ∣

\int \frac{- x - 1}{26 ( x ^{2} + 25 )} d x = \frac{1}{26} (- \frac{1}{2} ln x^{2} + 25 - \frac{1}{5} arctan (\frac{x}{5}))

= 26 (\frac{1}{26} ln ∣ x - 1 ∣ + \frac{1}{26} (- \frac{1}{2} ln x^{2} + 25 - \frac{1}{5} arctan (\frac{x}{5})))

Vereinfache

26 (\frac{1}{26} ln ∣ x - 1 ∣ + \frac{1}{26} (- \frac{1}{2} ln x^{2} + 25 - \frac{1}{5} arctan (\frac{x}{5}))) : ln ∣ x - 1 ∣ - \frac{1}{2} ln x^{2} + 25 - \frac{1}{5} arctan (\frac{x}{5})

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

Füge eine Konstante zur Lösung hinzu

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

\int \frac{2 x - 3}{4 x - x ^{2}} d x

\int_{e}^{e^{2}} \frac{1}{x ln ( x )} d x

\frac{d}{d x} (\frac{40}{1 + 79 e ^{- x}})

\frac{d}{d x} (\frac{x}{x - 7})

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