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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of ((2p+3)/((p-1)^2))

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

integral of (arcsin(sqrt(x)))/(sqrt(x(1-x)))

\int \frac{arcsin ( x )}{x ( 1 - x )} d x

integral of 3/(3x-5)

\int \frac{3}{3 x - 5} d x

integral of 4^{5x}

\int 4^{5 x} d x

integral of x/(sqrt(5-4x))

\int \frac{x}{5 - 4 x} d x