integral 1/((a^2+x^2)^{0.5)}

Lösung

integral

\frac{1}{( a ^{2} + x ^{2} ) ^{0.5}}

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

Schritte zur Lösung

\int \frac{1}{( a ^{2} + x ^{2} ) ^{0.5}} d x

Wende die partielle Integration an

= \frac{x}{( a ^{2} + x ^{2} ) ^{\frac{1}{2}}} - \int - \frac{x ^{2}}{( a ^{2} + x ^{2} ) ^{1.5}} d x

\int - \frac{x ^{2}}{( a ^{2} + x ^{2} ) ^{1.5}} d x = - ln (\frac{x + a ^{2} + x ^{2}}{∣ a ∣}) + \frac{x}{a ^{2} + x ^{2}}

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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

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

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

in t e g r a l \frac{1}{36 + x ^{2}}

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