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Beliebt Rechnen >

integral of (x^3)/(sqrt(49+9x^2))

Lösung

\int \frac{x ^{3}}{49 + 9 x ^{2}} d x

Lösung

\frac{1}{81} (\frac{1}{3} (49 + 9 x^{2})^{\frac{3}{2}} - 49 49 + 9 x^{2}) + C

Schritte zur Lösung

\int \frac{x ^{3}}{49 + 9 x ^{2}} d x

Wende U-Substitution an

= \int \frac{u ^{2} - 49}{81} d u

Entferne die Konstante:

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

= \frac{1}{81} \cdot \int u^{2} - 49 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}{81} (\int u^{2} d u - \int 49 d u)

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

\int 49 d u = 49 u

= \frac{1}{81} (\frac{u ^{3}}{3} - 49 u)

Setze in

u = 49 + 9 x^{2}

ein

= \frac{1}{81} (\frac{( 49 + 9 x ^{2} ) ^{3}}{3} - 49 49 + 9 x^{2})

Vereinfache

\frac{1}{81} (\frac{( 49 + 9 x ^{2} ) ^{3}}{3} - 49 49 + 9 x^{2}) : \frac{1}{81} (\frac{1}{3} (49 + 9 x^{2})^{\frac{3}{2}} - 49 49 + 9 x^{2})

= \frac{1}{81} (\frac{1}{3} (49 + 9 x^{2})^{\frac{3}{2}} - 49 49 + 9 x^{2})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{81} (\frac{1}{3} (49 + 9 x^{2})^{\frac{3}{2}} - 49 49 + 9 x^{2}) + C

Graph

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