integral of (8x^2+6)/((x-5)^3)

Lösung

\int \frac{8 x ^{2} + 6}{( x - 5 ) ^{3}} d x

8 ln ∣ x - 5 ∣ - \frac{80}{x - 5} - \frac{103}{( x - 5 ) ^{2}} + C

Schritte zur Lösung

\int \frac{8 x ^{2} + 6}{( x - 5 ) ^{3}} d x

Wende U-Substitution an

= \int \frac{8 u ^{2} + 80 u + 206}{u ^{3}} d u

Multipliziere aus

\frac{8 u ^{2} + 80 u + 206}{u ^{3}} : \frac{8}{u} + \frac{80}{u ^{2}} + \frac{206}{u ^{3}}

= \int \frac{8}{u} + \frac{80}{u ^{2}} + \frac{206}{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

= \int \frac{8}{u} d u + \int \frac{80}{u ^{2}} d u + \int \frac{206}{u ^{3}} d u

\int \frac{8}{u} d u = 8 ln ∣ u ∣

\int \frac{80}{u ^{2}} d u = - \frac{80}{u}

\int \frac{206}{u ^{3}} d u = - \frac{103}{u ^{2}}

= 8 ln ∣ u ∣ - \frac{80}{u} - \frac{103}{u ^{2}}

Setze in

u = x - 5

ein

= 8 ln ∣ x - 5 ∣ - \frac{80}{x - 5} - \frac{103}{( x - 5 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= 8 ln ∣ x - 5 ∣ - \frac{80}{x - 5} - \frac{103}{( x - 5 ) ^{2}} + C

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