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

Lösung

integral

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

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

= \frac{1}{2} (ln ∣ u ∣ - (- \frac{1}{u}))

Setze in

u = 1 + x^{2}

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l sin^{2} (x) + cos^{2} (x)

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

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

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

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