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Beliebt Rechnen >

integral of sqrt(8+3x^2)

Lösung

\int 8 + 3 x^{2} d x

Lösung

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

Schritte zur Lösung

\int 8 + 3 x^{2} d x

Trigonometrische Substitution anwenden

= \int \frac{8}{3} sec^{3} (u) d u

Entferne die Konstante:

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

= \frac{8}{3} \cdot \int sec^{3} (u) d u

Wende integrale Reduktion an

= \frac{8}{3} (\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{8}{3} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Setze in

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

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (2x+1)^7ln(x)

\int (2 x + 1)^{7} ln (x) d x

integral of (tan(2x+1))/(cos^2(2x+1))

\int \frac{tan ( 2 x + 1 )}{cos ^{2} ( 2 x + 1 )} d x

integral of (ln(2x))

\int (ln (2 x)) d x

integral of (2x^2-csc^2(ln(x)))/x

\int \frac{2 x ^{2} - csc ^{2} ( ln ( x ) )}{x} d x

integral of sec^8(θ)tan(θ)

\int sec^{8} (θ) tan (θ) d θ