integral of 1/(sqrt(x^2-8x+25))

Lösung

\int \frac{1}{x ^{2} - 8 x + 25} d x

ln (\frac{1}{3} x - 4 + x^{2} - 8 x + 25) + C

Schritte zur Lösung

\int \frac{1}{x ^{2} - 8 x + 25} d x

Vervollständige das Quadrat

x^{2} - 8 x + 25 : (x - 4)^{2} + 9

= \int \frac{1}{( x - 4 ) ^{2} + 9} d x

Wende U-Substitution an

= \int \frac{1}{u ^{2} + 9} d u

Trigonometrische Substitution anwenden

= \int sec (v) d v

Nutze das gemeinsame Integral :

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

= ln ∣ tan (v) + sec (v) ∣

Ersetze zurück

= ln tan (arctan (\frac{1}{3} (x - 4))) + sec (arctan (\frac{1}{3} (x - 4)))

Vereinfache

ln tan (arctan (\frac{1}{3} (x - 4))) + sec (arctan (\frac{1}{3} (x - 4))) : ln (\frac{1}{3} x - 4 + x^{2} - 8 x + 25)

= ln (\frac{1}{3} x - 4 + x^{2} - 8 x + 25)

Füge eine Konstante zur Lösung hinzu

= ln (\frac{1}{3} x - 4 + x^{2} - 8 x + 25) + C

\frac{d}{d t} (2 sin (3 t))

s l o p e (80, 2000), (260, 5600)

\int_{2}^{7} \frac{1}{25 + ( x - 2 ) ^{2}} d x

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