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

Lösung

integral

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

\frac{x}{( x ^{2} + a ) ^{2}} + \frac{1}{8 a ^{\frac{3}{2}}} (4 arctan (\frac{x}{a}) - sin (4 arctan (\frac{x}{a}))) + C

Schritte zur Lösung

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

Wende die partielle Integration an

= \frac{x}{( x ^{2} + a ) ^{2}} - \int - \frac{4 x ^{2}}{( x ^{2} + a ) ^{3}} d x

\int - \frac{4 x ^{2}}{( x ^{2} + a ) ^{3}} d x = - \frac{1}{8 a ^{\frac{3}{2}}} (4 arctan (\frac{x}{a}) - sin (4 arctan (\frac{x}{a})))

= \frac{x}{( x ^{2} + a ) ^{2}} - (- \frac{1}{8 a ^{\frac{3}{2}}} (4 arctan (\frac{x}{a}) - sin (4 arctan (\frac{x}{a}))))

Vereinfache

= \frac{x}{( x ^{2} + a ) ^{2}} + \frac{1}{8 a ^{\frac{3}{2}}} (4 arctan (\frac{x}{a}) - sin (4 arctan (\frac{x}{a})))

Füge eine Konstante zur Lösung hinzu

= \frac{x}{( x ^{2} + a ) ^{2}} + \frac{1}{8 a ^{\frac{3}{2}}} (4 arctan (\frac{x}{a}) - sin (4 arctan (\frac{x}{a}))) + C

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

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

in t e g r a l (1 + t^{2})^{- 1}

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

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