integral of ((7x+15y)/(x^2+y^2))

Lösung

\int (\frac{7 x + 15 y}{x ^{2} + y ^{2}}) d x

15 arctan (\frac{x}{y}) + \frac{7}{2} ln y^{2} + x^{2} + C

Schritte zur Lösung

\int \frac{7 x + 15 y}{x ^{2} + y ^{2}} d x

Multipliziere aus

\frac{15 y + 7 x}{y ^{2} + x ^{2}} : \frac{15 y}{y ^{2} + x ^{2}} + \frac{7 x}{y ^{2} + x ^{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{15 y}{y ^{2} + x ^{2}} d x + \int \frac{7 x}{y ^{2} + x ^{2}} d x

\int \frac{15 y}{y ^{2} + x ^{2}} d x = 15 arctan (\frac{x}{y})

\int \frac{7 x}{y ^{2} + x ^{2}} d x = \frac{7}{2} ln y^{2} + x^{2}

= 15 arctan (\frac{x}{y}) + \frac{7}{2} ln y^{2} + x^{2}

Füge eine Konstante zur Lösung hinzu

= 15 arctan (\frac{x}{y}) + \frac{7}{2} ln y^{2} + x^{2} + C