integral of (24)/((x-2)(x^2+4))

Lösung

\int \frac{24}{( x - 2 ) ( x ^{2} + 4 )} d x

6 (\frac{1}{2} ln \frac{x}{2} - 1 + \frac{1}{2} (- \frac{1}{2} ln \frac{x ^{2}}{4} + 1 - arctan (\frac{x}{2}))) + C

Schritte zur Lösung

\int \frac{24}{( x - 2 ) ( x ^{2} + 4 )} d x

Entferne die Konstante:

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

= 24 \cdot \int \frac{1}{( x - 2 ) ( x ^{2} + 4 )} d x

Wende integrale Substitution an

= 24 \cdot \int \frac{1}{4 ( u - 1 ) ( u ^{2} + 1 )} d u

Entferne die Konstante:

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

= 24 \cdot \frac{1}{4} \cdot \int \frac{1}{( u - 1 ) ( u ^{2} + 1 )} d u

Ermittle den Partialbruch von

\frac{1}{( u - 1 ) ( u ^{2} + 1 )} : \frac{1}{2 ( u - 1 )} + \frac{- u - 1}{2 ( u ^{2} + 1 )}

= 24 \cdot \frac{1}{4} \cdot \int \frac{1}{2 ( u - 1 )} + \frac{- u - 1}{2 ( u ^{2} + 1 )} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 24 \cdot \frac{1}{4} (\int \frac{1}{2 ( u - 1 )} d u + \int \frac{- u - 1}{2 ( u ^{2} + 1 )} d u)

\int \frac{1}{2 ( u - 1 )} d u = \frac{1}{2} ln ∣ u - 1 ∣

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

= 24 \cdot \frac{1}{4} (\frac{1}{2} ln ∣ u - 1 ∣ + \frac{1}{2} (- \frac{1}{2} ln u^{2} + 1 - arctan (u)))

Setze in

u = \frac{x}{2}

ein

= 24 \cdot \frac{1}{4} (\frac{1}{2} ln \frac{x}{2} - 1 + \frac{1}{2} (- \frac{1}{2} ln (\frac{x}{2})^{2} + 1 - arctan (\frac{x}{2})))

Vereinfache

24 \cdot \frac{1}{4} (\frac{1}{2} ln \frac{x}{2} - 1 + \frac{1}{2} (- \frac{1}{2} ln (\frac{x}{2})^{2} + 1 - arctan (\frac{x}{2}))) : 6 (\frac{1}{2} ln \frac{x}{2} - 1 + \frac{1}{2} (- \frac{1}{2} ln \frac{x ^{2}}{4} + 1 - arctan (\frac{x}{2})))

= 6 (\frac{1}{2} ln \frac{x}{2} - 1 + \frac{1}{2} (- \frac{1}{2} ln \frac{x ^{2}}{4} + 1 - arctan (\frac{x}{2})))

Füge eine Konstante zur Lösung hinzu

= 6 (\frac{1}{2} ln \frac{x}{2} - 1 + \frac{1}{2} (- \frac{1}{2} ln \frac{x ^{2}}{4} + 1 - arctan (\frac{x}{2}))) + C

\int_{0}^{1} \frac{x ^{2}}{x ^{3} - 1} d x

d er i v a t i v e y = e^{5 x}

ma c l a u r in ln (cos (x))

x \to 2 lim (x^{2} - 5 x + 6)

d er i v a t i v e (7 x^{2} - 12 x) e^{x}