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Beliebt Rechnen >

integral of sqrt(4t^2+1)

Lösung

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

Lösung

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

Schritte zur Lösung

\int 4 t^{2} + 1 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 0 of-|x-2|+3

x \to 0 lim (- ∣ x - 2 ∣ + 3)

(\partial)/(\partial x)((x+y)/(x*y))

\frac{\partial}{\partial x} (\frac{x + y}{x \cdot y})

derivative of log_{3}(4x)

\frac{d}{d x} (lo g_{3} (4 x))

d/(dθ)(8/(2+cot(θ)sin(θ)))

\frac{d}{d θ} (\frac{8}{2 + cot ( θ ) sin ( θ )})

integral of 1/(x(ln(x))^4)

\int \frac{1}{x ( ln ( x ) ) ^{4}} d x