integral of (5x^2)/((x-8)^3)

Lösung

\int \frac{5 x ^{2}}{( x - 8 ) ^{3}} d x

5 (ln ∣ x - 8 ∣ - \frac{16}{x - 8} - \frac{32}{( x - 8 ) ^{2}}) + C

Schritte zur Lösung

\int \frac{5 x ^{2}}{( x - 8 ) ^{3}} d x

Entferne die Konstante:

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

= 5 \cdot \int \frac{x ^{2}}{( x - 8 ) ^{3}} d x

Wende U-Substitution an

= 5 \cdot \int \frac{( u + 8 ) ^{2}}{u ^{3}} d u

Multipliziere aus

\frac{( u + 8 ) ^{2}}{u ^{3}} : \frac{1}{u} + \frac{16}{u ^{2}} + \frac{64}{u ^{3}}

= 5 \cdot \int \frac{1}{u} + \frac{16}{u ^{2}} + \frac{64}{u ^{3}} 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 (\int \frac{1}{u} d u + \int \frac{16}{u ^{2}} d u + \int \frac{64}{u ^{3}} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{16}{u ^{2}} d u = - \frac{16}{u}

\int \frac{64}{u ^{3}} d u = - \frac{32}{u ^{2}}

= 5 (ln ∣ u ∣ - \frac{16}{u} - \frac{32}{u ^{2}})

Setze in

u = x - 8

ein

= 5 (ln ∣ x - 8 ∣ - \frac{16}{x - 8} - \frac{32}{( x - 8 ) ^{2}})

Füge eine Konstante zur Lösung hinzu

= 5 (ln ∣ x - 8 ∣ - \frac{16}{x - 8} - \frac{32}{( x - 8 ) ^{2}}) + C

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