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Beliebt Rechnen >

integral of sqrt(2+4t^2)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Trigonometrische Substitution anwenden

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

Wende integrale Reduktion an

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

Setze in

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

ein

= \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

\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)) : \frac{1}{2} t 1 + 2 t^{2} + \frac{1}{2} ln 2 t + 1 + 2 t^{2}

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of x^33^x

\int x^{3} 3^{x} d x

integral of 5x^3+2x-6

\int 5 x^{3} + 2 x - 6 d x

integral of (x^4-a^4)/(x^2)

\int \frac{x ^{4} - a ^{4}}{x ^{2}} d x

integral of (8x^3+13x)/(x^4+4x^2+4)

\int \frac{8 x ^{3} + 13 x}{x ^{4} + 4 x ^{2} + 4} d x

in t e g r a l 9 x^{2}