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

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \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

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

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

\int 7 d u = 7 u

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

Setze in

u = x^{2} + 7

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e f (0.6) = 3 arcsin (x^{4})

\frac{\partial}{\partial x} (\frac{3 x ^{2}}{y - 5})

\frac{\partial}{\partial z} (\frac{x + y}{z})

t a y l or ln (1 - e^{- x})

\frac{d}{d x} (4 x^{2} + 3 x - 7)