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Beliebt Rechnen >

integral of sqrt(4t^2+4)

Lösung

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

Lösung

t 1 + t^{2} + ln t + 1 + t^{2} + C

Schritte zur Lösung

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

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

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

Wende integrale Reduktion an

= 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) ∣

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

Setze in

u = arctan (\frac{2}{2} t)

ein

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

Vereinfache

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

= t 1 + t^{2} + ln t + 1 + t^{2}

Füge eine Konstante zur Lösung hinzu

= t 1 + t^{2} + ln t + 1 + t^{2} + C

Graph

Beliebte Beispiele

integral of sin(2x+pi)

\int sin (2 x + π) d x

limit as x approaches 0 of xsin(1/x)

x \to 0 lim (x sin (\frac{1}{x}))

integral from 1 to 2 of (10ln(x))/(x^2)

\int_{1}^{2} \frac{10 ln ( x )}{x ^{2}} d x

f(x)=arctan(5x)

f (x) = arctan (5 x)

integral of x/(sqrt(x-11))

\int \frac{x}{x - 11} d x