integral of (ax+b)/(a^2x^2+b^2)

Lösung

\int \frac{a x + b}{a ^{2} x ^{2} + b ^{2}} d x

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

Schritte zur Lösung

\int \frac{a x + b}{a ^{2} x ^{2} + b ^{2}} d x

Multipliziere aus

\frac{a x + b}{a ^{2} x ^{2} + b ^{2}} : \frac{a x}{a ^{2} x ^{2} + b ^{2}} + \frac{b}{a ^{2} x ^{2} + b ^{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{a x}{a ^{2} x ^{2} + b ^{2}} d x + \int \frac{b}{a ^{2} x ^{2} + b ^{2}} d x

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

\int \frac{b}{a ^{2} x ^{2} + b ^{2}} d x = \frac{1}{a} arctan (\frac{a}{b} x)

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d y}{d x} + 4 y = 0

d er i v a t i v e f (x) = 3 x^{2} + 2 x - 3

y^{^{''}} - 8 y^{^{'}} = 0

\int 3 x e^{2 x} d x

y^{^{'}} = \frac{1}{2 y}