integral of (2x+5)/(x^2+4x+7)

Lösung

\int \frac{2 x + 5}{x ^{2} + 4 x + 7} d x

ln x^{2} + 4 x + 7 + \frac{1}{3} arctan (\frac{1}{3} (x + 2)) + C

Schritte zur Lösung

\int \frac{2 x + 5}{x ^{2} + 4 x + 7} d x

Vervollständige das Quadrat

x^{2} + 4 x + 7 : (x + 2)^{2} + 3

= \int \frac{2 x + 5}{( x + 2 ) ^{2} + 3} d x

Wende U-Substitution an

= \int \frac{2 u + 1}{u ^{2} + 3} d u

Multipliziere aus

\frac{2 u + 1}{u ^{2} + 3} : \frac{2 u}{u ^{2} + 3} + \frac{1}{u ^{2} + 3}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{2 u}{u ^{2} + 3} d u + \int \frac{1}{u ^{2} + 3} d u

\int \frac{2 u}{u ^{2} + 3} d u = ln u^{2} + 3

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

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

Setze in

u = x + 2

ein

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

Vereinfache

ln (x + 2)^{2} + 3 + \frac{1}{3} arctan (\frac{x + 2}{3}) : ln x^{2} + 4 x + 7 + \frac{1}{3} arctan (\frac{1}{3} (x + 2))

= ln x^{2} + 4 x + 7 + \frac{1}{3} arctan (\frac{1}{3} (x + 2))

Füge eine Konstante zur Lösung hinzu

= ln x^{2} + 4 x + 7 + \frac{1}{3} arctan (\frac{1}{3} (x + 2)) + C

in t e g r a l 5 x^{3}

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