integral of (x+y)/(x^2+y^2)

Lösung

\int \frac{x + y}{x ^{2} + y ^{2}} d y

arctan (\frac{y}{x}) + \frac{1}{2} ln x^{2} + y^{2} + C

Schritte zur Lösung

\int \frac{x + y}{x ^{2} + y ^{2}} d y

Multipliziere aus

\frac{x + y}{x ^{2} + y ^{2}} : \frac{x}{x ^{2} + y ^{2}} + \frac{y}{x ^{2} + y ^{2}}

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} + y ^{2}} d y + \int \frac{y}{x ^{2} + y ^{2}} d y

\int \frac{x}{x ^{2} + y ^{2}} d y = arctan (\frac{y}{x})

\int \frac{y}{x ^{2} + y ^{2}} d y = \frac{1}{2} ln x^{2} + y^{2}

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

Füge eine Konstante zur Lösung hinzu

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

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