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integral of sqrt(3x^2+4x-7)

Lösung

\int 3 x^{2} + 4 x - 7 d x

\frac{1}{6 3} ((3 x + 2) 9 x^{2} + 12 x - 21 - 25 ln (\frac{3 x + 9 x ^{2} + 12 x - 21 + 2}{5})) + C

Schritte zur Lösung

\int 3 x^{2} + 4 x - 7 d x

Vervollständige das Quadrat

3 x^{2} + 4 x - 7 : 3 (x + \frac{2}{3})^{2} - \frac{25}{3}

= \int 3 (x + \frac{2}{3})^{2} - \frac{25}{3} d x

Wende U-Substitution an

= \int \frac{9 u ^{2} - 25}{3} d u

\frac{9 u ^{2} - 25}{3} = \frac{1}{3} 9 u^{2} - 25

= \int \frac{1}{3} 9 u^{2} - 25 d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int 9 u^{2} - 25 d u

Trigonometrische Substitution anwenden

= \frac{1}{3} \cdot \int \frac{25}{3} tan^{2} (v) sec (v) d v

Entferne die Konstante:

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

= \frac{1}{3} \frac{25}{3} \cdot \int tan^{2} (v) sec (v) d v

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{1}{3} \frac{25}{3} \cdot \int (- 1 + sec^{2} (v)) sec (v) d v

Multipliziere aus

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

= \frac{1}{3} \frac{25}{3} \cdot \int - sec (v) + sec^{3} (v) d v

Wende die Summenregel an:

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

= \frac{1}{3} \frac{25}{3} (- \int sec (v) d v + \int sec^{3} (v) d v)

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

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

= \frac{1}{3} \frac{25}{3} (- ln ∣ tan (v) + sec (v) ∣ + \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Ersetze zurück

= \frac{1}{3} \frac{25}{3} (- ln tan (arcsec (\frac{3}{5} (x + \frac{2}{3}))) + sec (arcsec (\frac{3}{5} (x + \frac{2}{3}))) + \frac{1}{2} sec (arcsec (\frac{3}{5} (x + \frac{2}{3}))) tan (arcsec (\frac{3}{5} (x + \frac{2}{3}))) + \frac{1}{2} ln tan (arcsec (\frac{3}{5} (x + \frac{2}{3}))) + sec (arcsec (\frac{3}{5} (x + \frac{2}{3}))))

Vereinfache

\frac{1}{3} \frac{25}{3} (- ln tan (arcsec (\frac{3}{5} (x + \frac{2}{3}))) + sec (arcsec (\frac{3}{5} (x + \frac{2}{3}))) + \frac{1}{2} sec (arcsec (\frac{3}{5} (x + \frac{2}{3}))) tan (arcsec (\frac{3}{5} (x + \frac{2}{3}))) + \frac{1}{2} ln tan (arcsec (\frac{3}{5} (x + \frac{2}{3}))) + sec (arcsec (\frac{3}{5} (x + \frac{2}{3})))) : \frac{1}{6 3} ((3 x + 2) 9 x^{2} + 12 x - 21 - 25 ln (\frac{3 x + 9 x ^{2} + 12 x - 21 + 2}{5}))

= \frac{1}{6 3} ((3 x + 2) 9 x^{2} + 12 x - 21 - 25 ln (\frac{3 x + 9 x ^{2} + 12 x - 21 + 2}{5}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{6 3} ((3 x + 2) 9 x^{2} + 12 x - 21 - 25 ln (\frac{3 x + 9 x ^{2} + 12 x - 21 + 2}{5})) + C

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