What is the integral of (ax+b)/(a^2x^2+b^2) ?

The integral of (ax+b)/(a^2x^2+b^2) is 1/(2a)ln|a^2x^2+b^2|+1/a arctan(a/b x)+C

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

\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

Solution steps

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

Expand

\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}}

Apply the Sum Rule:

\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)

Add a constant to the solution

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

What is the integral of (ax+b)/(a^2x^2+b^2) ?
The integral of (ax+b)/(a^2x^2+b^2) is 1/(2a)ln|a^2x^2+b^2|+1/a arctan(a/b x)+C