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Beliebt Rechnen >

integral of 3/(sqrt(x^2+16))

Lösung

\int \frac{3}{x ^{2} + 16} d x

Lösung

3 ln (\frac{1}{4} x + 16 + x^{2}) + C

Schritte zur Lösung

\int \frac{3}{x ^{2} + 16} d x

Entferne die Konstante:

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

= 3 \cdot \int \frac{1}{x ^{2} + 16} d x

Trigonometrische Substitution anwenden

= 3 \cdot \int sec (u) d u

Nutze das gemeinsame Integral :

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

= 3 ln ∣ tan (u) + sec (u) ∣

Setze in

u = arctan (\frac{1}{4} x)

ein

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

Vereinfache

3 ln tan (arctan (\frac{1}{4} x)) + sec (arctan (\frac{1}{4} x)) : 3 ln (\frac{1}{4} x + 16 + x^{2})

= 3 ln (\frac{1}{4} x + 16 + x^{2})

Füge eine Konstante zur Lösung hinzu

= 3 ln (\frac{1}{4} x + 16 + x^{2}) + C

Graph

Beliebte Beispiele

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\int 5 x^{- 4} d x

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t an g e n t f (x) = \frac{3}{x}, at x = 6

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\frac{d}{d x} (5 ln (cos (x)))

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integral from 1 to 12 of 1/(x-10)

\int_{1}^{12} \frac{1}{x - 10} d x