integral of 1/(sqrt(x^2+2x+122))

Lösung

\int \frac{1}{x ^{2} + 2 x + 122} d x

ln (\frac{1}{11} x + 1 + x^{2} + 2 x + 122) + C

Schritte zur Lösung

\int \frac{1}{x ^{2} + 2 x + 122} d x

Vervollständige das Quadrat

x^{2} + 2 x + 122 : (x + 1)^{2} + 121

= \int \frac{1}{( x + 1 ) ^{2} + 121} d x

Wende U-Substitution an

= \int \frac{1}{u ^{2} + 121} 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}{11} (x + 1))) + sec (arctan (\frac{1}{11} (x + 1)))

Vereinfache

ln tan (arctan (\frac{1}{11} (x + 1))) + sec (arctan (\frac{1}{11} (x + 1))) : ln (\frac{1}{11} x + 1 + x^{2} + 2 x + 122)

= ln (\frac{1}{11} x + 1 + x^{2} + 2 x + 122)

Füge eine Konstante zur Lösung hinzu

= ln (\frac{1}{11} x + 1 + x^{2} + 2 x + 122) + C

l c m \frac{csc ^{2} ( x )}{4}, 4 sin^{2} (x)

\int x^{\frac{5}{2}} d x

ma c l a u r in \frac{( e ^{x} - e ^{- x} )}{2}

t an g e n t \frac{4}{x}

t an g e n t y = x, at x = 81