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

Lösung

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

4 (ln ∣ x - 8 ∣ - \frac{8}{x - 8}) + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende U-Substitution an

= 4 \cdot \int \frac{u + 8}{u ^{2}} d u

Multipliziere aus

\frac{u + 8}{u ^{2}} : \frac{1}{u} + \frac{8}{u ^{2}}

= 4 \cdot \int \frac{1}{u} + \frac{8}{u ^{2}} 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

= 4 (\int \frac{1}{u} d u + \int \frac{8}{u ^{2}} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{8}{u ^{2}} d u = - \frac{8}{u}

= 4 (ln ∣ u ∣ - \frac{8}{u})

Setze in

u = x - 8

ein

= 4 (ln ∣ x - 8 ∣ - \frac{8}{x - 8})

Füge eine Konstante zur Lösung hinzu

= 4 (ln ∣ x - 8 ∣ - \frac{8}{x - 8}) + C

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