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

Lösung

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

\frac{1}{5} (x^{2} + 3)^{\frac{5}{2}} - (x^{2} + 3)^{\frac{3}{2}} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

(u^{2} - 3) u^{2} : u^{4} - 3 u^{2}

= \int u^{4} - 3 u^{2} 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^{4} d u - \int 3 u^{2} d u

\int u^{4} d u = \frac{u ^{5}}{5}

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

= \frac{u ^{5}}{5} - u^{3}

Setze in

u = x^{2} + 3

ein

= \frac{( x ^{2} + 3 ) ^{5}}{5} - (x^{2} + 3)^{3}

Vereinfache

\frac{( x ^{2} + 3 ) ^{5}}{5} - (x^{2} + 3)^{3} : \frac{1}{5} (x^{2} + 3)^{\frac{5}{2}} - (x^{2} + 3)^{\frac{3}{2}}

= \frac{1}{5} (x^{2} + 3)^{\frac{5}{2}} - (x^{2} + 3)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5} (x^{2} + 3)^{\frac{5}{2}} - (x^{2} + 3)^{\frac{3}{2}} + C

d er i v a t i v e f (x) = 5 e^{4 x}

x \frac{d y}{d x} + 2 y = - 3 e^{x}

\int 2 x^{- 4} d x

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

y^{^{''}} - 4 y^{^{'}} + 5 y = e^{2 x}