integral of ((x^3))/(sqrt(1+x^2))

Lösung

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

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

Schritte zur Lösung

\int \frac{x ^{3}}{1 + x ^{2}} d x

Wende U-Substitution an

= \int u^{2} - 1 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 1 d u

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

\int 1 d u = u

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

Setze in

u = 1 + x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e 2 x^{2} + x - 2

\int_{0}^{\infty} \frac{5}{x ^{3}} d x

t an g e n t 0 = x^{2} - x y + 2 y - 2, at x = - 2

\frac{\partial}{\partial x} (x^{3} \cdot y^{2} + 4 x^{2} - 3 x y)

x \to \infty lim (x - {x})