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Beliebt Rechnen >

integral of sqrt(x^2+3/4)

Lösung

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

Lösung

\frac{1}{8} (2 x 4 x^{2} + 3 + 3 (ln 2 x + 3 \frac{4 x ^{2} + 3}{3} - \frac{ln ( 3 )}{2})) + C

Schritte zur Lösung

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

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

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

Wende integrale Reduktion an

= \frac{3}{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) ∣

= \frac{3}{4} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Setze in

u = arctan (\frac{1}{\frac{3}{2}} x)

ein

= \frac{3}{4} \frac{sec ^{2} ( arctan ( \frac{1}{\frac{3}{2}} x ) ) sin ( arctan ( \frac{1}{\frac{3}{2}} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{\frac{3}{2}} x)) + sec (arctan (\frac{1}{\frac{3}{2}} x))

Vereinfache

\frac{3}{4} \frac{sec ^{2} ( arctan ( \frac{1}{\frac{3}{2}} x ) ) sin ( arctan ( \frac{1}{\frac{3}{2}} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{\frac{3}{2}} x)) + sec (arctan (\frac{1}{\frac{3}{2}} x)) : \frac{1}{8} (2 x 4 x^{2} + 3 + 3 (ln 2 x + 3 \frac{4 x ^{2} + 3}{3} - \frac{ln ( 3 )}{2}))

= \frac{1}{8} (2 x 4 x^{2} + 3 + 3 (ln 2 x + 3 \frac{4 x ^{2} + 3}{3} - \frac{ln ( 3 )}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} (2 x 4 x^{2} + 3 + 3 (ln 2 x + 3 \frac{4 x ^{2} + 3}{3} - \frac{ln ( 3 )}{2})) + C

Graph

Beliebte Beispiele

integral of ysin(y^2)

\int y sin (y^{2}) d y

integral of (24x)/(\sqrt[3]{3x^2-4)}

\int \frac{24 x}{3 3 x ^{2} - 4} d x

integral of 5sin^2(xco)s^2x

\int 5 sin^{2} (x co) s^{2} x d x

y^{''}+9y=18sin(3t)

y^{^{''}} + 9 y = 18 sin (3 t)

limit as x approaches 0+of sin(x^x)

x \to 0 + lim (sin (x^{x}))