integral of (11x^3)/(sqrt(9+x^2))

Lösung

\int \frac{11 x ^{3}}{9 + x ^{2}} d x

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

Schritte zur Lösung

\int \frac{11 x ^{3}}{9 + x ^{2}} d x

Entferne die Konstante:

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

= 11 \cdot \int \frac{x ^{3}}{9 + x ^{2}} d x

Wende U-Substitution an

= 11 \cdot \int u^{2} - 9 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

= 11 (\int u^{2} d u - \int 9 d u)

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

\int 9 d u = 9 u

= 11 (\frac{u ^{3}}{3} - 9 u)

Setze in

u = 9 + x^{2}

ein

= 11 (\frac{( 9 + x ^{2} ) ^{3}}{3} - 9 9 + x^{2})

Vereinfache

11 (\frac{( 9 + x ^{2} ) ^{3}}{3} - 9 9 + x^{2}) : 11 (\frac{1}{3} (9 + x^{2})^{\frac{3}{2}} - 9 9 + x^{2})

= 11 (\frac{1}{3} (9 + x^{2})^{\frac{3}{2}} - 9 9 + x^{2})

Füge eine Konstante zur Lösung hinzu

= 11 (\frac{1}{3} (9 + x^{2})^{\frac{3}{2}} - 9 9 + x^{2}) + C

\int \frac{1}{( x - 3 ) ( x - 2 )} d x

\frac{\partial}{\partial x} (2 x^{2} - 5 y^{2})

\int_{- 2}^{6} 4 x - x^{2} + 12 d x

y^{^{''}} + 5 y^{^{'}} = e^{- t}, y (0) = 0, y^{^{'}} (0) = 1

x \to 4 lim (f (x))