integral of (1+x^2)^{-2}

Lösung

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

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

Schritte zur Lösung

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

Wende Exponentenregel an:

a^{- b} = \frac{1}{a ^{b}}

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

Trigonometrische Substitution anwenden

= \int \frac{1}{sec ^{4} ( u ) cos ^{2} ( u )} d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{1 + cos ( 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 1 + cos (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 cos (2 u) d u)

\int 1 d u = u

\int cos (2 u) d u = \frac{1}{2} sin (2 u)

= \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Setze in

u = arctan (x)

ein

= \frac{1}{2} (arctan (x) + \frac{1}{2} sin (2 arctan (x)))

Vereinfache

\frac{1}{2} (arctan (x) + \frac{1}{2} sin (2 arctan (x))) : \frac{1}{2} (arctan (x) + \frac{x}{1 + x ^{2}})

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

Füge eine Konstante zur Lösung hinzu

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

(x^{6})^{^{'}}

\frac{d}{d x} (9 + x^{2})

\frac{d}{d x} (e^{x^{4} y})

\int x^{3} (5 + x^{4})^{11} d x

\int_{x^{3}}^{x^{4}} (2 t - 1)^{3} d t