integral (x^3)/((x^2+a^2))

Lösung

integral

\frac{x ^{3}}{( x ^{2} + a ^{2} )}

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{u - a ^{2}}{2 u} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{u - a ^{2}}{u} d u

Multipliziere aus

\frac{u - a ^{2}}{u} : 1 - \frac{a ^{2}}{u}

= \frac{1}{2} \cdot \int 1 - \frac{a ^{2}}{u} 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

= \frac{1}{2} (\int 1 d u - \int \frac{a ^{2}}{u} d u)

\int 1 d u = u

\int \frac{a ^{2}}{u} d u = a^{2} ln ∣ u ∣

= \frac{1}{2} (u - a^{2} ln ∣ u ∣)

Setze in

u = x^{2} + a^{2}

ein

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l \frac{e ^{- x}}{10}

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

in t e g r a l 2 y sin (x) - 4

in t e g r a l \frac{1}{( v ^{2} + 3 )}

in t e g r a l 4^{3 x}