integral of 5/((x-2)(x^2+4))

Lösung

\int \frac{5}{( x - 2 ) ( x ^{2} + 4 )} d x

\frac{5}{4} (\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{5}{( x - 2 ) ( x ^{2} + 4 )} d x

Entferne die Konstante:

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

= 5 \cdot \int \frac{1}{( x - 2 ) ( x ^{2} + 4 )} d x

Wende integrale Substitution an

= 5 \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

= 5 \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 )}

= 5 \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

= 5 \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))

= 5 \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

= 5 \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

5 \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}))) : \frac{5}{4} (\frac{1}{2} ln \frac{x}{2} - 1 + \frac{1}{2} (- \frac{1}{2} ln \frac{x ^{2}}{4} + 1 - arctan (\frac{x}{2})))

= \frac{5}{4} (\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

= \frac{5}{4} (\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 x^{2} I n d x

d er i v a t i v e lo g_{a} (x)

n = 3 \sum \infty \frac{2}{n ^{2} - 2 n}

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

x \to - \infty lim (e^{3 x})