integral of x^2sqrt(25+x^2)

Lösung

\int x^{2} 25 + x^{2} d x

625 (- \frac{1}{200} (x 25 + x^{2} + 25 ln (\frac{x + 25 + x ^{2}}{5})) + \frac{1}{2500} x (x^{2} + 25)^{\frac{3}{2}}) + C

Schritte zur Lösung

\int x^{2} 25 + x^{2} d x

Trigonometrische Substitution anwenden

= \int 625 sec^{3} (u) tan^{2} (u) d u

Entferne die Konstante:

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

= 625 \cdot \int sec^{3} (u) tan^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 625 \cdot \int sec^{3} (u) (- 1 + sec^{2} (u)) d u

Multipliziere aus

sec^{3} (u) (- 1 + sec^{2} (u)) : - sec^{3} (u) + sec^{5} (u)

= 625 \cdot \int - sec^{3} (u) + sec^{5} (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

= 625 (- \int sec^{3} (u) d u + \int sec^{5} (u) d u)

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

\int sec^{5} (u) d u = \frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

= 625 (- (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣) + \frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣))

Setze in

u = arctan (\frac{1}{5} x)

ein

= 625 (- (\frac{1}{2} sec (arctan (\frac{1}{5} x)) tan (arctan (\frac{1}{5} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x))) + \frac{sec ^{4} ( arctan ( \frac{1}{5} x ) ) sin ( arctan ( \frac{1}{5} x ) )}{4} + \frac{3}{4} (\frac{1}{2} sec (arctan (\frac{1}{5} x)) tan (arctan (\frac{1}{5} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x))))

Vereinfache

625 (- (\frac{1}{2} sec (arctan (\frac{1}{5} x)) tan (arctan (\frac{1}{5} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x))) + \frac{sec ^{4} ( arctan ( \frac{1}{5} x ) ) sin ( arctan ( \frac{1}{5} x ) )}{4} + \frac{3}{4} (\frac{1}{2} sec (arctan (\frac{1}{5} x)) tan (arctan (\frac{1}{5} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x)))) : 625 (- \frac{1}{200} (x 25 + x^{2} + 25 ln (\frac{x + 25 + x ^{2}}{5})) + \frac{1}{2500} x (x^{2} + 25)^{\frac{3}{2}})

= 625 (- \frac{1}{200} (x 25 + x^{2} + 25 ln (\frac{x + 25 + x ^{2}}{5})) + \frac{1}{2500} x (x^{2} + 25)^{\frac{3}{2}})

Füge eine Konstante zur Lösung hinzu

= 625 (- \frac{1}{200} (x 25 + x^{2} + 25 ln (\frac{x + 25 + x ^{2}}{5})) + \frac{1}{2500} x (x^{2} + 25)^{\frac{3}{2}}) + C

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