What is the integral of 4/x-(4x)/(x^2+1) ?

The integral of 4/x-(4x)/(x^2+1) is 4ln|x|-2ln|x^2+1|+C

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integral of 4/x-(4x)/(x^2+1)

\int \frac{4}{x} - \frac{4 x}{x ^{2} + 1} d x

4 ln ∣ x ∣ - 2 ln x^{2} + 1 + C

Solution steps

\int \frac{4}{x} - \frac{4 x}{x ^{2} + 1} d x

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{4}{x} d x - \int \frac{4 x}{x ^{2} + 1} d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

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

= 4 ln ∣ x ∣ - 2 ln x^{2} + 1

Add a constant to the solution

= 4 ln ∣ x ∣ - 2 ln x^{2} + 1 + C

What is the integral of 4/x-(4x)/(x^2+1) ?
The integral of 4/x-(4x)/(x^2+1) is 4ln|x|-2ln|x^2+1|+C