integral of (-8)/((x-2)(x^2+4))

Lösung

\int \frac{- 8}{( x - 2 ) ( x ^{2} + 4 )} d x

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

Schritte zur Lösung

\int \frac{- 8}{( x - 2 ) ( x ^{2} + 4 )} d x

Entferne die Konstante:

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

= - 8 \cdot \int \frac{1}{( x - 2 ) ( x ^{2} + 4 )} d x

Wende integrale Substitution an

= - 8 \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

= - 8 \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 )}

= - 8 \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

= - 8 \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))

= - 8 \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

= - 8 \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

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

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

Füge eine Konstante zur Lösung hinzu

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

- 3 y^{^{'}} + 9 x^{2} = 0

\frac{\partial}{\partial x} (6 y^{2} x)

(x x)^{^{'}}

\frac{d y}{d t} + 7 y = e^{4 t}, y (0) = 10

\frac{\partial}{\partial x} (e^{- a x^{2}})