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integral of sqrt(36x^2+12x+10)

Lösung

\int 36 x^{2} + 12 x + 10 d x

\frac{1}{12} (6 x 36 x^{2} + 12 x + 10 + 36 x^{2} + 12 x + 10 + 9 ln (\frac{6 ( x + \frac{1}{6} ) + 36 x ^{2} + 12 x + 10}{3})) + C

Schritte zur Lösung

\int 36 x^{2} + 12 x + 10 d x

36 x^{2} + 12 x + 10 = 2 18 x^{2} + 6 x + 5

= \int 2 18 x^{2} + 6 x + 5 d x

Entferne die Konstante:

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

= 2 \cdot \int 18 x^{2} + 6 x + 5 d x

Vervollständige das Quadrat

18 x^{2} + 6 x + 5 : 18 (x + \frac{1}{6})^{2} + \frac{9}{2}

= 2 \cdot \int 18 (x + \frac{1}{6})^{2} + \frac{9}{2} d x

Wende U-Substitution an

= 2 \cdot \int \frac{36 u ^{2} + 9}{2} d u

\frac{36 u ^{2} + 9}{2} = \frac{9}{2} 4 u^{2} + 1

= 2 \cdot \int \frac{9}{2} 4 u^{2} + 1 d u

Entferne die Konstante:

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

= 2 \frac{9}{2} \cdot \int 4 u^{2} + 1 d u

\frac{9}{2} = \frac{3}{2}

= 2 \frac{3}{2} \cdot \int 4 u^{2} + 1 d u

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

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

Wende integrale Reduktion an

= 2 \frac{3}{2} \cdot \frac{1}{2} (\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) ∣

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

Ersetze zurück

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

Vereinfache

2 \frac{3}{2} \cdot \frac{1}{2} (\frac{sec ^{2} ( arctan ( 2 ( x + \frac{1}{6} ) ) ) sin ( arctan ( 2 ( x + \frac{1}{6} ) ) )}{2} + \frac{1}{2} ln tan (arctan (2 (x + \frac{1}{6}))) + sec (arctan (2 (x + \frac{1}{6})))) : \frac{1}{12} (6 x 36 x^{2} + 12 x + 10 + 36 x^{2} + 12 x + 10 + 9 ln (\frac{6 ( x + \frac{1}{6} ) + 36 x ^{2} + 12 x + 10}{3}))

= \frac{1}{12} (6 x 36 x^{2} + 12 x + 10 + 36 x^{2} + 12 x + 10 + 9 ln (\frac{6 ( x + \frac{1}{6} ) + 36 x ^{2} + 12 x + 10}{3}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{12} (6 x 36 x^{2} + 12 x + 10 + 36 x^{2} + 12 x + 10 + 9 ln (\frac{6 ( x + \frac{1}{6} ) + 36 x ^{2} + 12 x + 10}{3})) + C

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