integral of sqrt(3x^2+5x+7)

Lösung

\int 3 x^{2} + 5 x + 7 d x

\frac{1}{24 3} ((6 x + 5) 36 x^{2} + 60 x + 84 + 59 (ln 6 x + 59 \frac{36 x ^{2} + 60 x + 84}{59} + 5 - \frac{ln ( 59 )}{2})) + C

Schritte zur Lösung

\int 3 x^{2} + 5 x + 7 d x

Vervollständige das Quadrat

3 x^{2} + 5 x + 7 : 3 (x + \frac{5}{6})^{2} + \frac{59}{12}

= \int 3 (x + \frac{5}{6})^{2} + \frac{59}{12} d x

Wende U-Substitution an

= \int \frac{36 u ^{2} + 59}{2 3} d u

Entferne die Konstante:

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

= \frac{1}{2 3} \cdot \int 36 u^{2} + 59 d u

Trigonometrische Substitution anwenden

= \frac{1}{2 3} \cdot \int \frac{59}{6} sec^{3} (v) d v

Entferne die Konstante:

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

= \frac{1}{2 3} \cdot \frac{59}{6} \cdot \int sec^{3} (v) d v

Wende integrale Reduktion an

= \frac{1}{2 3} \cdot \frac{59}{6} (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} \cdot \int sec (v) d v)

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

= \frac{1}{2 3} \cdot \frac{59}{6} (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

Ersetze zurück

= \frac{1}{2 3} \cdot \frac{59}{6} \frac{sec ^{2} ( arctan ( \frac{6}{59} ( x + \frac{5}{6} ) ) ) sin ( arctan ( \frac{6}{59} ( x + \frac{5}{6} ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{6}{59} (x + \frac{5}{6}))) + sec (arctan (\frac{6}{59} (x + \frac{5}{6})))

Vereinfache

\frac{1}{2 3} \cdot \frac{59}{6} \frac{sec ^{2} ( arctan ( \frac{6}{59} ( x + \frac{5}{6} ) ) ) sin ( arctan ( \frac{6}{59} ( x + \frac{5}{6} ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{6}{59} (x + \frac{5}{6}))) + sec (arctan (\frac{6}{59} (x + \frac{5}{6}))) : \frac{1}{24 3} ((6 x + 5) 36 x^{2} + 60 x + 84 + 59 (ln 6 x + 59 \frac{36 x ^{2} + 60 x + 84}{59} + 5 - \frac{ln ( 59 )}{2}))

= \frac{1}{24 3} ((6 x + 5) 36 x^{2} + 60 x + 84 + 59 (ln 6 x + 59 \frac{36 x ^{2} + 60 x + 84}{59} + 5 - \frac{ln ( 59 )}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{24 3} ((6 x + 5) 36 x^{2} + 60 x + 84 + 59 (ln 6 x + 59 \frac{36 x ^{2} + 60 x + 84}{59} + 5 - \frac{ln ( 59 )}{2})) + C

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