integral of (7x^3)/(sqrt(7+x^2))

Lösung

\int \frac{7 x ^{3}}{7 + x ^{2}} d x

7 (\frac{1}{3} (7 + x^{2})^{\frac{3}{2}} - 7 7 + x^{2}) + C

Schritte zur Lösung

\int \frac{7 x ^{3}}{7 + x ^{2}} d x

Entferne die Konstante:

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

= 7 \cdot \int \frac{x ^{3}}{7 + x ^{2}} d x

Wende U-Substitution an

= 7 \cdot \int u^{2} - 7 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

= 7 (\int u^{2} d u - \int 7 d u)

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

\int 7 d u = 7 u

= 7 (\frac{u ^{3}}{3} - 7 u)

Setze in

u = 7 + x^{2}

ein

= 7 (\frac{( 7 + x ^{2} ) ^{3}}{3} - 7 7 + x^{2})

Vereinfache

7 (\frac{( 7 + x ^{2} ) ^{3}}{3} - 7 7 + x^{2}) : 7 (\frac{1}{3} (7 + x^{2})^{\frac{3}{2}} - 7 7 + x^{2})

= 7 (\frac{1}{3} (7 + x^{2})^{\frac{3}{2}} - 7 7 + x^{2})

Füge eine Konstante zur Lösung hinzu

= 7 (\frac{1}{3} (7 + x^{2})^{\frac{3}{2}} - 7 7 + x^{2}) + C

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