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Beliebt Rechnen >

integral of sqrt(16+x^2)

Lösung

\int 16 + x^{2} d x

Lösung

16 (\frac{1}{32} x 16 + x^{2} + \frac{1}{2} ln (\frac{x + 16 + x ^{2}}{4})) + C

Schritte zur Lösung

\int 16 + x^{2} d x

Trigonometrische Substitution anwenden

= \int 16 sec^{3} (u) d u

Entferne die Konstante:

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

= 16 \cdot \int sec^{3} (u) d u

Wende integrale Reduktion an

= 16 (\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) ∣

= 16 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Setze in

u = arctan (\frac{1}{4} x)

ein

= 16 (\frac{sec ^{2} ( arctan ( \frac{1}{4} x ) ) sin ( arctan ( \frac{1}{4} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{4} x)) + sec (arctan (\frac{1}{4} x)))

Vereinfache

16 (\frac{sec ^{2} ( arctan ( \frac{1}{4} x ) ) sin ( arctan ( \frac{1}{4} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{4} x)) + sec (arctan (\frac{1}{4} x))) : 16 (\frac{1}{32} x 16 + x^{2} + \frac{1}{2} ln (\frac{x + 16 + x ^{2}}{4}))

= 16 (\frac{1}{32} x 16 + x^{2} + \frac{1}{2} ln (\frac{x + 16 + x ^{2}}{4}))

Füge eine Konstante zur Lösung hinzu

= 16 (\frac{1}{32} x 16 + x^{2} + \frac{1}{2} ln (\frac{x + 16 + x ^{2}}{4})) + C

Graph

Beliebte Beispiele

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steigung (-45)(-4-8)

s l o p e (- 45) (- 4 - 8)

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