integral of (x^5)/(x^2sqrt(9x^2+9))

Lösung

\int \frac{x ^{5}}{x ^{2} 9 x ^{2} + 9} d x

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

Schritte zur Lösung

\int \frac{x ^{5}}{x ^{2} 9 x ^{2} + 9} d x

Vereinfache

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

= \int \frac{x ^{3}}{3 x ^{2} + 1} d x

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int \frac{x ^{3}}{x ^{2} + 1} d x

Wende U-Substitution an

= \frac{1}{3} \cdot \int u^{2} - 1 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

= \frac{1}{3} (\int u^{2} d u - \int 1 d u)

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

\int 1 d u = u

= \frac{1}{3} (\frac{u ^{3}}{3} - u)

Setze in

u = x^{2} + 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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