integral of (3x^3)/(sqrt(x^2+16))

Lösung

\int \frac{3 x ^{3}}{x ^{2} + 16} d x

(x^{2} + 16)^{\frac{3}{2}} - 48 x^{2} + 16 + C

Schritte zur Lösung

\int \frac{3 x ^{3}}{x ^{2} + 16} d x

Entferne die Konstante:

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

= 3 \cdot \int \frac{x ^{3}}{x ^{2} + 16} d x

Wende U-Substitution an

= 3 \cdot \int 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

= 3 (\int u^{2} d u - \int 16 d u)

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

\int 16 d u = 16 u

= 3 (\frac{u ^{3}}{3} - 16 u)

Setze in

u = x^{2} + 16

ein

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

Vereinfache

3 (\frac{( x ^{2} + 16 ) ^{3}}{3} - 16 x^{2} + 16) : (x^{2} + 16)^{\frac{3}{2}} - 48 x^{2} + 16

= (x^{2} + 16)^{\frac{3}{2}} - 48 x^{2} + 16

Füge eine Konstante zur Lösung hinzu

= (x^{2} + 16)^{\frac{3}{2}} - 48 x^{2} + 16 + C

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