integral ((x^3+2))/((4-x^2))

Lösung

integral

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

- \frac{x ^{2}}{2} - 2 ln - x^{2} + 4 + \frac{1}{2} ln \frac{x}{2} + 1 - \frac{1}{2} ln \frac{x}{2} - 1 + C

Schritte zur Lösung

\int \frac{x ^{3} + 2}{4 - x ^{2}} d x

Multipliziere aus

\frac{x ^{3} + 2}{4 - x ^{2}} : \frac{x ^{3}}{4 - x ^{2}} + \frac{2}{4 - x ^{2}}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{3}}{4 - x ^{2}} d x + \int \frac{2}{4 - x ^{2}} d x

\int \frac{x ^{3}}{4 - x ^{2}} d x = - \frac{x ^{2}}{2} - 2 ln - x^{2} + 4

\int \frac{2}{4 - x ^{2}} d x = \frac{1}{2} ln \frac{x}{2} + 1 - \frac{1}{2} ln \frac{x}{2} - 1

= - \frac{x ^{2}}{2} - 2 ln - x^{2} + 4 + \frac{1}{2} ln \frac{x}{2} + 1 - \frac{1}{2} ln \frac{x}{2} - 1

Füge eine Konstante zur Lösung hinzu

= - \frac{x ^{2}}{2} - 2 ln - x^{2} + 4 + \frac{1}{2} ln \frac{x}{2} + 1 - \frac{1}{2} ln \frac{x}{2} - 1 + C

in t e g r a l x^{2} (x - 4)^{\frac{1}{2}}

in t e g r a l x^{2} \cdot (e^{x})^{2}

in t e g r a l e^{x} - 2^{x}

in t e g r a l \frac{x}{( x ^{2} - 1 )}

in t e g r a l \frac{1}{( c - x )}