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Beliebt Rechnen >

integral of (3x+1)/(x^2-10x+28)

Lösung

\int \frac{3 x + 1}{x ^{2} - 10 x + 28} d x

Lösung

\frac{3}{2} ln x^{2} - 10 x + 28 + \frac{16}{3} arctan (\frac{1}{3} (x - 5)) + C

Schritte zur Lösung

\int \frac{3 x + 1}{x ^{2} - 10 x + 28} d x

Vervollständige das Quadrat

x^{2} - 10 x + 28 : (x - 5)^{2} + 3

= \int \frac{3 x + 1}{( x - 5 ) ^{2} + 3} d x

Wende U-Substitution an

= \int \frac{3 u + 16}{u ^{2} + 3} d u

Multipliziere aus

\frac{3 u + 16}{u ^{2} + 3} : \frac{3 u}{u ^{2} + 3} + \frac{16}{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{3 u}{u ^{2} + 3} d u + \int \frac{16}{u ^{2} + 3} d u

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

\int \frac{16}{u ^{2} + 3} d u = \frac{16}{3} arctan (\frac{u}{3})

= \frac{3}{2} ln u^{2} + 3 + \frac{16}{3} arctan (\frac{u}{3})

Setze in

u = x - 5

ein

= \frac{3}{2} ln (x - 5)^{2} + 3 + \frac{16}{3} arctan (\frac{x - 5}{3})

Vereinfache

\frac{3}{2} ln (x - 5)^{2} + 3 + \frac{16}{3} arctan (\frac{x - 5}{3}) : \frac{3}{2} ln x^{2} - 10 x + 28 + \frac{16}{3} arctan (\frac{1}{3} (x - 5))

= \frac{3}{2} ln x^{2} - 10 x + 28 + \frac{16}{3} arctan (\frac{1}{3} (x - 5))

Füge eine Konstante zur Lösung hinzu

= \frac{3}{2} ln x^{2} - 10 x + 28 + \frac{16}{3} arctan (\frac{1}{3} (x - 5)) + C

Graph

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