integral of (x+2)/(sqrt(x^2-6x+10))

Lösung

\int \frac{x + 2}{x ^{2} - 6 x + 10} d x

x^{2} - 6 x + 10 + 5 ln x^{2} - 6 x + 10 + x - 3 + C

Schritte zur Lösung

\int \frac{x + 2}{x ^{2} - 6 x + 10} d x

Multipliziere aus

\frac{x + 2}{x ^{2} - 6 x + 10} : \frac{x}{x ^{2} - 6 x + 10} + \frac{2}{x ^{2} - 6 x + 10}

= \int \frac{x}{x ^{2} - 6 x + 10} + \frac{2}{x ^{2} - 6 x + 10} d x

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{x}{x ^{2} - 6 x + 10} d x + \int \frac{2}{x ^{2} - 6 x + 10} d x

\int \frac{x}{x ^{2} - 6 x + 10} d x = x^{2} - 6 x + 10 + 3 ln x^{2} - 6 x + 10 + x - 3

\int \frac{2}{x ^{2} - 6 x + 10} d x = 2 ln x^{2} - 6 x + 10 + x - 3

= x^{2} - 6 x + 10 + 3 ln x^{2} - 6 x + 10 + x - 3 + 2 ln x^{2} - 6 x + 10 + x - 3

Vereinfache

= x^{2} - 6 x + 10 + 5 ln x^{2} - 6 x + 10 + x - 3

Füge eine Konstante zur Lösung hinzu

= x^{2} - 6 x + 10 + 5 ln x^{2} - 6 x + 10 + x - 3 + C

\int \frac{6 x ^{2}}{5 4 x ^{3} - 3} d x

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