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Beliebt Rechnen >

integral of sqrt(1+2x^2)

Lösung

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

Lösung

\frac{1}{2} (\frac{1}{2} x 1 + 2 x^{2} + \frac{1}{2} ln 2 x + 1 + 2 x^{2}) + C

Schritte zur Lösung

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

Trigonometrische Substitution anwenden

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

Wende integrale Reduktion an

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

Setze in

u = arctan (2 x)

ein

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

Vereinfache

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

= \frac{1}{2} (\frac{1}{2} x 1 + 2 x^{2} + \frac{1}{2} ln 2 x + 1 + 2 x^{2})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (\frac{1}{2} x 1 + 2 x^{2} + \frac{1}{2} ln 2 x + 1 + 2 x^{2}) + C

Graph

Beliebte Beispiele

integral of (x-2)/(x^2-10x+106)

\int \frac{x - 2}{x ^{2} - 10 x + 106} d x

integral of sec(2y)tan(2y)

\int sec (2 y) tan (2 y) d y

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\int 9 (x^{2} + 3 x + 5) (2 x + 3) d x

integral of (ln^2(x))/(sqrt(x^5))

\int \frac{ln ^{2} ( x )}{x ^{5}} d x

integral of (x^5-x^3)

\int (x^{5} - x^{3}) d x