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

Lösung

\int \frac{x ^{2}}{x - 4} d x

\frac{( x - 4 ) ^{2}}{2} + 8 (x - 4) + 16 ln ∣ x - 4 ∣ + C

Schritte zur Lösung

\int \frac{x ^{2}}{x - 4} d x

Wende U-Substitution an

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

Multipliziere aus

\frac{( u + 4 ) ^{2}}{u} : u + 8 + \frac{16}{u}

= \int u + 8 + \frac{16}{u} 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 u d u + \int 8 d u + \int \frac{16}{u} d u

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

\int 8 d u = 8 u

\int \frac{16}{u} d u = 16 ln ∣ u ∣

= \frac{u ^{2}}{2} + 8 u + 16 ln ∣ u ∣

Setze in

u = x - 4

ein

= \frac{( x - 4 ) ^{2}}{2} + 8 (x - 4) + 16 ln ∣ x - 4 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{( x - 4 ) ^{2}}{2} + 8 (x - 4) + 16 ln ∣ x - 4 ∣ + C

d er i v a t i v e 5

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x \to 0 lim ((1 - 7 x)^{\frac{1}{x}})