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Beliebt Rechnen >

integral of sqrt(1+4t^2)

Lösung

\int 1 + 4 t^{2} d t

Lösung

\frac{1}{4} (2 t 4 t^{2} + 1 + ln 2 t + 1 + 4 t^{2}) + C

Schritte zur Lösung

\int 1 + 4 t^{2} d t

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 t)

ein

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

Vereinfache

\frac{1}{2} (\frac{sec ^{2} ( arctan ( 2 t ) ) sin ( arctan ( 2 t ) )}{2} + \frac{1}{2} ln ∣ tan (arctan (2 t)) + sec (arctan (2 t)) ∣) : \frac{1}{4} (2 t 4 t^{2} + 1 + ln 2 t + 1 + 4 t^{2})

= \frac{1}{4} (2 t 4 t^{2} + 1 + ln 2 t + 1 + 4 t^{2})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} (2 t 4 t^{2} + 1 + ln 2 t + 1 + 4 t^{2}) + C

Graph

Beliebte Beispiele

limit as x approaches 2+of (-5)/(x-2)

x \to 2 + lim (\frac{- 5}{x - 2})

derivative of (2x-3/(2x+5))

\frac{d}{d x} (\frac{2 x - 3}{2 x + 5})

integral of (x-3)*cos(x)

\int (x - 3) \cdot cos (x) d x

integral of (5r^2)/(r+3)

\int \frac{5 r ^{2}}{r + 3} d r

derivative of x^2ln(5x)

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