integral of x^2(x^2+6)^{-1/2}

Lösung

\int x^{2} (x^{2} + 6)^{- \frac{1}{2}} d x

6 (- \frac{1}{2} ln \frac{1}{6 ^{\frac{1}{2}}} x + \frac{1}{6} (6 + x^{2}) + \frac{1}{12} x (6 + x^{2})^{\frac{1}{2}}) + C

Schritte zur Lösung

\int x^{2} (x^{2} + 6)^{- \frac{1}{2}} d x

Trigonometrische Substitution anwenden

= \int 6 tan^{2} (u) sec (u) d u

Entferne die Konstante:

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

= 6 \cdot \int tan^{2} (u) sec (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 6 \cdot \int (- 1 + sec^{2} (u)) sec (u) d u

Multipliziere aus

(- 1 + sec^{2} (u)) sec (u) : - sec (u) + sec^{3} (u)

= 6 \cdot \int - sec (u) + sec^{3} (u) d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 6 (- \int sec (u) d u + \int sec^{3} (u) d u)

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

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

= 6 (- ln ∣ tan (u) + sec (u) ∣ + \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Setze in

u = arctan (\frac{1}{6 ^{\frac{1}{2}}} x)

ein

= 6 (- ln tan (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + sec (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + \frac{1}{2} sec (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) tan (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + sec (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)))

Vereinfache

6 (- ln tan (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + sec (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + \frac{1}{2} sec (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) tan (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + sec (arctan (\frac{1}{6 ^{\frac{1}{2}}} x))) : 6 (- \frac{1}{2} ln \frac{1}{6 ^{\frac{1}{2}}} x + \frac{1}{6} (6 + x^{2}) + \frac{1}{12} x (6 + x^{2})^{\frac{1}{2}})

= 6 (- \frac{1}{2} ln \frac{1}{6 ^{\frac{1}{2}}} x + \frac{1}{6} (6 + x^{2}) + \frac{1}{12} x (6 + x^{2})^{\frac{1}{2}})

Füge eine Konstante zur Lösung hinzu

= 6 (- \frac{1}{2} ln \frac{1}{6 ^{\frac{1}{2}}} x + \frac{1}{6} (6 + x^{2}) + \frac{1}{12} x (6 + x^{2})^{\frac{1}{2}}) + C

\int \frac{48}{1 + 36 x ^{2}} d x

\int 8 x (4 x^{2} + 9)^{4} d x

\int \frac{x ^{2} + 8}{x ^{2} - 5 x + 6} d x

\int e^{- a x} cos (w x) d x

\int \frac{1}{x ln ( a )} d x