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Beliebt Rechnen >

integral of 1/(sqrt(4x^2-7))

Lösung

\int \frac{1}{4 x ^{2} - 7} d x

Lösung

\frac{1}{2} ln \frac{1}{7} (4 x^{2} - 7) + \frac{2}{7} x + C

Schritte zur Lösung

\int \frac{1}{4 x ^{2} - 7} d x

Trigonometrische Substitution anwenden

= \int \frac{sec ( u )}{2} d u

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

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

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

Setze in

u = arcsec (\frac{2}{7} x)

ein

= \frac{1}{2} ln tan (arcsec (\frac{2}{7} x)) + sec (arcsec (\frac{2}{7} x))

Vereinfache

\frac{1}{2} ln tan (arcsec (\frac{2}{7} x)) + sec (arcsec (\frac{2}{7} x)) : \frac{1}{2} ln \frac{1}{7} (4 x^{2} - 7) + \frac{2}{7} x

= \frac{1}{2} ln \frac{1}{7} (4 x^{2} - 7) + \frac{2}{7} x

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln \frac{1}{7} (4 x^{2} - 7) + \frac{2}{7} x + C

Graph

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