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Beliebt Rechnen >

integral of sqrt(x^2+21)

Lösung

\int x^{2} + 21 d x

Lösung

21 (\frac{1}{42} x 21 + x^{2} + \frac{1}{2} ln \frac{1}{21} x + \frac{1}{21} (21 + x^{2})) + C

Schritte zur Lösung

\int x^{2} + 21 d x

Trigonometrische Substitution anwenden

= \int 21 sec^{3} (u) d u

Entferne die Konstante:

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

= 21 \cdot \int sec^{3} (u) d u

Wende integrale Reduktion an

= 21 (\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) ∣

= 21 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Setze in

u = arctan (\frac{1}{21} x)

ein

= 21 \frac{sec ^{2} ( arctan ( \frac{1}{21} x ) ) sin ( arctan ( \frac{1}{21} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{21} x)) + sec (arctan (\frac{1}{21} x))

Vereinfache

21 \frac{sec ^{2} ( arctan ( \frac{1}{21} x ) ) sin ( arctan ( \frac{1}{21} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{21} x)) + sec (arctan (\frac{1}{21} x)) : 21 (\frac{1}{42} x 21 + x^{2} + \frac{1}{2} ln \frac{1}{21} x + \frac{1}{21} (21 + x^{2}))

= 21 (\frac{1}{42} x 21 + x^{2} + \frac{1}{2} ln \frac{1}{21} x + \frac{1}{21} (21 + x^{2}))

Füge eine Konstante zur Lösung hinzu

= 21 (\frac{1}{42} x 21 + x^{2} + \frac{1}{2} ln \frac{1}{21} x + \frac{1}{21} (21 + x^{2})) + C

Graph

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