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Beliebt Rechnen >

integral of sqrt(1+8x^2)

Lösung

\int 1 + 8 x^{2} d x

Lösung

\frac{1}{2} x 8 x^{2} + 1 + \frac{1}{4 2} ln 22 x + 1 + 8 x^{2} + C

Schritte zur Lösung

\int 1 + 8 x^{2} d x

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

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

Wende integrale Reduktion an

= \frac{1}{2 2} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

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

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

Setze in

u = arctan (22 x)

ein

= \frac{1}{2 2} (\frac{sec ^{2} ( arctan ( 2 2 x ) ) sin ( arctan ( 2 2 x ) )}{2} + \frac{1}{2} ln tan (arctan (22 x)) + sec (arctan (22 x)))

Vereinfache

\frac{1}{2 2} (\frac{sec ^{2} ( arctan ( 2 2 x ) ) sin ( arctan ( 2 2 x ) )}{2} + \frac{1}{2} ln tan (arctan (22 x)) + sec (arctan (22 x))) : \frac{1}{2} x 8 x^{2} + 1 + \frac{1}{4 2} ln 22 x + 1 + 8 x^{2}

= \frac{1}{2} x 8 x^{2} + 1 + \frac{1}{4 2} ln 22 x + 1 + 8 x^{2}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x 8 x^{2} + 1 + \frac{1}{4 2} ln 22 x + 1 + 8 x^{2} + C

Graph

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