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Beliebt Rechnen >

integral of (x^2)/(sqrt(x+4))

Lösung

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

Lösung

2 (\frac{1}{5} (x + 4)^{\frac{5}{2}} - \frac{8}{3} (x + 4)^{\frac{3}{2}} + 16 x + 4) + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

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

= 2 \cdot \int u^{4} - 8 u^{2} + 16 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 8 u^{2} d u + \int 16 d u)

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

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

\int 16 d u = 16 u

= 2 (\frac{u ^{5}}{5} - \frac{8 u ^{3}}{3} + 16 u)

Setze in

u = x + 4

ein

= 2 (\frac{( x + 4 ) ^{5}}{5} - \frac{8 ( x + 4 ) ^{3}}{3} + 16 x + 4)

Vereinfache

2 (\frac{( x + 4 ) ^{5}}{5} - \frac{8 ( x + 4 ) ^{3}}{3} + 16 x + 4) : 2 (\frac{1}{5} (x + 4)^{\frac{5}{2}} - \frac{8}{3} (x + 4)^{\frac{3}{2}} + 16 x + 4)

= 2 (\frac{1}{5} (x + 4)^{\frac{5}{2}} - \frac{8}{3} (x + 4)^{\frac{3}{2}} + 16 x + 4)

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{5} (x + 4)^{\frac{5}{2}} - \frac{8}{3} (x + 4)^{\frac{3}{2}} + 16 x + 4) + C

Graph

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