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Beliebt Rechnen >

integral of (sqrt(a^2+u^2))/u

Lösung

\int \frac{a ^{2} + u ^{2}}{u}

Lösung

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

Schritte zur Lösung

\int \frac{a ^{2} + u ^{2}}{u} d a

Entferne die Konstante:

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

= \frac{1}{u} \cdot \int u^{2} + a^{2} d a

Trigonometrische Substitution anwenden

= \frac{1}{u} \cdot \int u^{2} sec^{3} (v) d v

Entferne die Konstante:

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

= \frac{1}{u} u^{2} \cdot \int sec^{3} (v) d v

Wende integrale Reduktion an

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

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

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

Setze in

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

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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