What is the integral of 2/x+(3-x)/(x^2+1) ?

The integral of 2/x+(3-x)/(x^2+1) is 2ln|x|+3arctan(x)-1/2 ln|x^2+1|+C

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integral of 2/x+(3-x)/(x^2+1)

\int \frac{2}{x} + \frac{3 - x}{x ^{2} + 1} d x

2 ln ∣ x ∣ + 3 arctan (x) - \frac{1}{2} ln x^{2} + 1 + C

Solution steps

\int \frac{2}{x} + \frac{3 - 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{2}{x} d x + \int \frac{3 - x}{x ^{2} + 1} d x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{3 - x}{x ^{2} + 1} d x = 3 arctan (x) - \frac{1}{2} ln x^{2} + 1

= 2 ln ∣ x ∣ + 3 arctan (x) - \frac{1}{2} ln x^{2} + 1

Add a constant to the solution

= 2 ln ∣ x ∣ + 3 arctan (x) - \frac{1}{2} ln x^{2} + 1 + C

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\int tan (z) d z

What is the integral of 2/x+(3-x)/(x^2+1) ?
The integral of 2/x+(3-x)/(x^2+1) is 2ln|x|+3arctan(x)-1/2 ln|x^2+1|+C