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Beliebt Rechnen >

integral of sqrt(((x^2+4)))

Lösung

\int ((x^{2} + 4)) d x

Lösung

4 (\frac{1}{8} x 4 + x^{2} + \frac{1}{2} ln (\frac{x + 4 + x ^{2}}{2})) + C

Schritte zur Lösung

\int x^{2} + 4 d x

Trigonometrische Substitution anwenden

= \int 4 sec^{3} (u) d u

Entferne die Konstante:

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

= 4 \cdot \int sec^{3} (u) d u

Wende integrale Reduktion an

= 4 (\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) ∣

= 4 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Setze in

u = arctan (\frac{1}{2} x)

ein

= 4 (\frac{sec ^{2} ( arctan ( \frac{1}{2} x ) ) sin ( arctan ( \frac{1}{2} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{2} x)) + sec (arctan (\frac{1}{2} x)))

Vereinfache

4 (\frac{sec ^{2} ( arctan ( \frac{1}{2} x ) ) sin ( arctan ( \frac{1}{2} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{2} x)) + sec (arctan (\frac{1}{2} x))) : 4 (\frac{1}{8} x 4 + x^{2} + \frac{1}{2} ln (\frac{x + 4 + x ^{2}}{2}))

= 4 (\frac{1}{8} x 4 + x^{2} + \frac{1}{2} ln (\frac{x + 4 + x ^{2}}{2}))

Füge eine Konstante zur Lösung hinzu

= 4 (\frac{1}{8} x 4 + x^{2} + \frac{1}{2} ln (\frac{x + 4 + x ^{2}}{2})) + C

Graph

Beliebte Beispiele

limit as x approaches-5 of 12x+1

x \to - 5 lim (12 x + 1)

derivative h(x)=(x+x^{-1})^3

d er i v a t i v e h (x) = (x + x^{- 1})^{3}

integral of e^{4x+9}

\int e^{4 x + 9} d x

derivative of 4x(x^2-2)

\frac{d}{d x} (4 x (x^{2} - 2))

derivative f(x)=sin(cos(7x))

d er i v a t i v e f (x) = sin (cos (7 x))