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Beliebt Rechnen >

integral of 2arctan(x+2)

Lösung

\int 2 arctan (x + 2) d x

Lösung

2 x arctan (x + 2) + 4 arctan (x + 2) - ln x^{2} + 4 x + 5 + C

Schritte zur Lösung

\int 2 arctan (x + 2) d x

Entferne die Konstante:

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

= 2 \cdot \int arctan (x + 2) d x

Wende U-Substitution an

= 2 \cdot \int arctan (u) d u

Wende die partielle Integration an

= 2 (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

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

Setze in

u = x + 2

ein

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

Vereinfache

2 ((x + 2) arctan (x + 2) - \frac{1}{2} ln (x + 2)^{2} + 1) : 2 x arctan (x + 2) + 4 arctan (x + 2) - ln x^{2} + 4 x + 5

= 2 x arctan (x + 2) + 4 arctan (x + 2) - ln x^{2} + 4 x + 5

Füge eine Konstante zur Lösung hinzu

= 2 x arctan (x + 2) + 4 arctan (x + 2) - ln x^{2} + 4 x + 5 + C

Graph

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