integral of (x^2)/(sqrt(x-3))

Lösung

\int \frac{x ^{2}}{x - 3} d x

2 (\frac{1}{5} (x - 3)^{\frac{5}{2}} + 2 (x - 3)^{\frac{3}{2}} + 9 x - 3) + C

Schritte zur Lösung

\int \frac{x ^{2}}{x - 3} d x

Wende U-Substitution an

= \int 2 (u^{2} + 3)^{2} d u

Entferne die Konstante:

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

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

Multipliziere aus

(u^{2} + 3)^{2} : u^{4} + 6 u^{2} + 9

= 2 \cdot \int u^{4} + 6 u^{2} + 9 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

= 2 (\int u^{4} d u + \int 6 u^{2} d u + \int 9 d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int 6 u^{2} d u = 2 u^{3}

\int 9 d u = 9 u

= 2 (\frac{u ^{5}}{5} + 2 u^{3} + 9 u)

Setze in

u = x - 3

ein

= 2 (\frac{( x - 3 ) ^{5}}{5} + 2 (x - 3)^{3} + 9 x - 3)

Vereinfache

2 (\frac{( x - 3 ) ^{5}}{5} + 2 (x - 3)^{3} + 9 x - 3) : 2 (\frac{1}{5} (x - 3)^{\frac{5}{2}} + 2 (x - 3)^{\frac{3}{2}} + 9 x - 3)

= 2 (\frac{1}{5} (x - 3)^{\frac{5}{2}} + 2 (x - 3)^{\frac{3}{2}} + 9 x - 3)

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{5} (x - 3)^{\frac{5}{2}} + 2 (x - 3)^{\frac{3}{2}} + 9 x - 3) + C

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