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Beliebt Rechnen >

integral of sqrt(4+16x^2)

Lösung

\int 4 + 16 x^{2} d x

Lösung

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

Schritte zur Lösung

\int 4 + 16 x^{2} d x

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{4}{2} x)

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral from 1 to 9 of y^{1/2}ln(y)

\int_{1}^{9} y^{\frac{1}{2}} ln (y) d y

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\int \frac{v}{( v - 1 ) ^{2}} d v

t^2*(dy)/(dt)+y=0

t^{2} \cdot \frac{d y}{d t} + y = 0

integral of x(x-21)^2

\int x (x - 21)^{2} d x

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\int 3 sec (x) tan (x) d x