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

Lösung

integral

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

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

Schritte zur Lösung

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

Trigonometrische Substitution anwenden

= \int a^{2} sec^{3} (u) d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= a^{2} \cdot \int sec^{3} (u) d u

Wende integrale Reduktion an

= a^{2} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= a^{2} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Setze in

u = arctan (\frac{1}{a} x)

ein

= a^{2} (\frac{sec ^{2} ( arctan ( \frac{1}{a} x ) ) sin ( arctan ( \frac{1}{a} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{a} x)) + sec (arctan (\frac{1}{a} x)))

Vereinfache

a^{2} (\frac{sec ^{2} ( arctan ( \frac{1}{a} x ) ) sin ( arctan ( \frac{1}{a} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{a} x)) + sec (arctan (\frac{1}{a} x))) : \frac{1}{2} x (x^{2} + a^{2})^{\frac{1}{2}} + \frac{a ^{2}}{2} ln \frac{x + ( a ^{2} + x ^{2} ) ^{\frac{1}{2}}}{∣ a ∣}

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l 2 + 5 x

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

in t e g r a l e^{1 - x}

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

in t e g r a l \frac{( 6 x - 4 )}{( x ^{3} - 4 x )}