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Beliebt Rechnen >

integral of arctan(3x-1)

Lösung

\int arctan (3 x - 1) d x

Lösung

\frac{1}{6} (2 arctan (3 x - 1) (3 x - 1) - ln 9 x^{2} - 6 x + 2) + C

Schritte zur Lösung

\int arctan (3 x - 1) d x

Wende U-Substitution an

= \int arctan (u) \frac{1}{3} d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int arctan (u) d u

Wende die partielle Integration an

= \frac{1}{3} (u arctan (u) - \int \frac{u}{u ^{2} + 1} d u)

\int \frac{u}{u ^{2} + 1} d u = \frac{1}{2} ln u^{2} + 1

= \frac{1}{3} (u arctan (u) - \frac{1}{2} ln u^{2} + 1)

Setze in

u = 3 x - 1

ein

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

Vereinfache

\frac{1}{3} ((3 x - 1) arctan (3 x - 1) - \frac{1}{2} ln (3 x - 1)^{2} + 1) : \frac{1}{6} (2 arctan (3 x - 1) (3 x - 1) - ln 9 x^{2} - 6 x + 2)

= \frac{1}{6} (2 arctan (3 x - 1) (3 x - 1) - ln 9 x^{2} - 6 x + 2)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{6} (2 arctan (3 x - 1) (3 x - 1) - ln 9 x^{2} - 6 x + 2) + C

Graph

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