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Beliebt Rechnen >

integral of 1/(sqrt(x^2+4x+8))

Lösung

\int \frac{1}{x ^{2} + 4 x + 8} d x

Lösung

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

Schritte zur Lösung

\int \frac{1}{x ^{2} + 4 x + 8} d x

Vervollständige das Quadrat

x^{2} + 4 x + 8 : (x + 2)^{2} + 4

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

Wende U-Substitution an

= \int \frac{1}{u ^{2} + 4} 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}{2} (x + 2))) + sec (arctan (\frac{1}{2} (x + 2)))

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

5e^{2x}(dy)/(dx)=-10 x/(y^2)

5 e^{2 x} \frac{d y}{d x} = - 10 \frac{x}{y ^{2}}

tangent f(x)= 1/(2sqrt(x)),\at x=49

t an g e n t f (x) = \frac{1}{2 x}, at x = 49

derivative f(x)=9-4/(x^2)

d er i v a t i v e f (x) = 9 - \frac{4}{x ^{2}}

derivative of x^2sec(1/x)

\frac{d}{d x} (x^{2} sec (\frac{1}{x}))

integral of x^{3.4}-2x^{sqrt(2)-1}

\int x^{3.4} - 2 x^{2 - 1} d x