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Beliebt Rechnen >

integral of sqrt(r^2+1)

Lösung

\int r^{2} + 1 d r

Lösung

\frac{1}{2} r 1 + r^{2} + \frac{1}{2} ln r + 1 + r^{2} + C

Schritte zur Lösung

\int r^{2} + 1 d r

Trigonometrische Substitution anwenden

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

Wende integrale Reduktion an

= \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) ∣

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

Setze in

u = arctan (r)

ein

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

Vereinfache

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

= \frac{1}{2} r 1 + r^{2} + \frac{1}{2} ln r + 1 + r^{2}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} r 1 + r^{2} + \frac{1}{2} ln r + 1 + r^{2} + C

Graph

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