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Beliebt Rechnen >

integral of (x^2+12)/(xsqrt(x^2-9))

Lösung

\int \frac{x ^{2} + 12}{x x ^{2} - 9} d x

Lösung

x^{2} - 9 + 4 arctan (\frac{1}{3} x^{2} - 9) + C

Schritte zur Lösung

\int \frac{x ^{2} + 12}{x x ^{2} - 9} d x

Wende U-Substitution an

= \int \frac{u + 12}{2 u u - 9} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{u + 12}{u u - 9} d u

Multipliziere aus

\frac{u + 12}{u u - 9} : \frac{1}{u - 9} + \frac{12}{u u - 9}

= \frac{1}{2} \cdot \int \frac{1}{u - 9} + \frac{12}{u u - 9} d u

Wende die Summenregel an:

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

= \frac{1}{2} (\int \frac{1}{u - 9} d u + \int \frac{12}{u u - 9} d u)

\int \frac{1}{u - 9} d u = 2 u - 9

\int \frac{12}{u u - 9} d u = 8 arctan (\frac{1}{3} u - 9)

= \frac{1}{2} (2 u - 9 + 8 arctan (\frac{1}{3} u - 9))

Setze in

u = x^{2}

ein

= \frac{1}{2} (2 x^{2} - 9 + 8 arctan (\frac{1}{3} x^{2} - 9))

Vereinfache

\frac{1}{2} (2 x^{2} - 9 + 8 arctan (\frac{1}{3} x^{2} - 9)) : x^{2} - 9 + 4 arctan (\frac{1}{3} x^{2} - 9)

= x^{2} - 9 + 4 arctan (\frac{1}{3} x^{2} - 9)

Füge eine Konstante zur Lösung hinzu

= x^{2} - 9 + 4 arctan (\frac{1}{3} x^{2} - 9) + C

Graph

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