integral of (x+3)/(sqrt(x^2+4))

Lösung

\int \frac{x + 3}{x ^{2} + 4} d x

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

Schritte zur Lösung

\int \frac{x + 3}{x ^{2} + 4} d x

Multipliziere aus

\frac{x + 3}{x ^{2} + 4} : \frac{x}{x ^{2} + 4} + \frac{3}{x ^{2} + 4}

= \int \frac{x}{x ^{2} + 4} + \frac{3}{x ^{2} + 4} 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} + 4} d x + \int \frac{3}{x ^{2} + 4} d x

\int \frac{x}{x ^{2} + 4} d x = x^{2} + 4

\int \frac{3}{x ^{2} + 4} d x = 3 ln (\frac{1}{2} x + 4 + x^{2})

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

Füge eine Konstante zur Lösung hinzu

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

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