integral of (5x+12)/(x^2+4x+5)

Lösung

\int \frac{5 x + 12}{x ^{2} + 4 x + 5} d x

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

Schritte zur Lösung

\int \frac{5 x + 12}{x ^{2} + 4 x + 5} d x

Vervollständige das Quadrat

x^{2} + 4 x + 5 : (x + 2)^{2} + 1

= \int \frac{5 x + 12}{( x + 2 ) ^{2} + 1} d x

Wende U-Substitution an

= \int \frac{5 u + 2}{u ^{2} + 1} d u

Multipliziere aus

\frac{5 u + 2}{u ^{2} + 1} : \frac{5 u}{u ^{2} + 1} + \frac{2}{u ^{2} + 1}

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{5 u}{u ^{2} + 1} d u + \int \frac{2}{u ^{2} + 1} d u

\int \frac{5 u}{u ^{2} + 1} d u = \frac{5}{2} ln u^{2} + 1

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

= \frac{5}{2} ln u^{2} + 1 + 2 arctan (u)

Setze in

u = x + 2

ein

= \frac{5}{2} ln (x + 2)^{2} + 1 + 2 arctan (x + 2)

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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