integral of sqrt(5+3x^2)

Lösung

\int 5 + 3 x^{2} d x

\frac{1}{2} x 3 x^{2} + 5 + \frac{5}{2 3} ln \frac{3}{5} x + \frac{1}{5} (5 + 3 x^{2}) + C

Schritte zur Lösung

\int 5 + 3 x^{2} d x

Trigonometrische Substitution anwenden

= \int \frac{5}{3} sec^{3} (u) d u

Entferne die Konstante:

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

= \frac{5}{3} \cdot \int sec^{3} (u) d u

Wende integrale Reduktion an

= \frac{5}{3} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

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

= \frac{5}{3} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Setze in

u = arctan (\frac{3}{5} x)

ein

= \frac{5}{3} \frac{sec ^{2} ( arctan ( \frac{3}{5} x ) ) sin ( arctan ( \frac{3}{5} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{3}{5} x)) + sec (arctan (\frac{3}{5} x))

Vereinfache

\frac{5}{3} \frac{sec ^{2} ( arctan ( \frac{3}{5} x ) ) sin ( arctan ( \frac{3}{5} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{3}{5} x)) + sec (arctan (\frac{3}{5} x)) : \frac{1}{2} x 3 x^{2} + 5 + \frac{5}{2 3} ln \frac{3}{5} x + \frac{1}{5} (5 + 3 x^{2})

= \frac{1}{2} x 3 x^{2} + 5 + \frac{5}{2 3} ln \frac{3}{5} x + \frac{1}{5} (5 + 3 x^{2})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x 3 x^{2} + 5 + \frac{5}{2 3} ln \frac{3}{5} x + \frac{1}{5} (5 + 3 x^{2}) + C

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

\int ln (s) d s

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

in t e g r a l x^{-} 1

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