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

Lösung

integral

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

ln ∣ x ∣ + 2 arctan (x) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ + 2 arctan (x)

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ + 2 arctan (x) + C

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

in t e g r a l 5^{2 x - 7}

in t e g r a l \frac{1}{3 x ^{2} - 3 x}

in t e g r a l \frac{( x ^{2} + 4 )}{( x + 3 )}

in t e g r a l x^{5} (x^{3} + 4)^{\frac{1}{2}}