de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of sqrt(3x^2+2)

Lösung

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

Lösung

\frac{1}{2} x 3 x^{2} + 2 + \frac{1}{3} ln \frac{3}{2} x + \frac{2 + 3 x ^{2}}{2} + C

Schritte zur Lösung

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

Trigonometrische Substitution anwenden

= \int \frac{2}{3} sec^{3} (u) d u

Entferne die Konstante:

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

= \frac{2}{3} \cdot \int sec^{3} (u) d u

Wende integrale Reduktion an

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

Setze in

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

ein

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

Vereinfache

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

= \frac{1}{2} x 3 x^{2} + 2 + \frac{1}{3} ln \frac{3}{2} x + \frac{2 + 3 x ^{2}}{2}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x 3 x^{2} + 2 + \frac{1}{3} ln \frac{3}{2} x + \frac{2 + 3 x ^{2}}{2} + C

Graph

Beliebte Beispiele

integral of (x^2+6x+4)/(x^4+8x^2+16)

\int \frac{x ^{2} + 6 x + 4}{x ^{4} + 8 x ^{2} + 16} d x

integral of 9/(xln(10))

\int \frac{9}{x ln ( 10 )} d x

integral of 1/((y^2-4))

\int \frac{1}{( y ^{2} - 4 )} d y

integral of 1/(1+sqrt(1-z^2))

\int \frac{1}{1 + 1 - z ^{2}} d z

integral of ((6+4x))/(1+x^2)

\int \frac{( 6 + 4 x )}{1 + x ^{2}} d x