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Beliebt Rechnen >

integral of sqrt(2+4x^2)

Lösung

\int 2 + 4 x^{2} d x

Lösung

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

Schritte zur Lösung

\int 2 + 4 x^{2} d x

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 (\frac{2}{2} x)

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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