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

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

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

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

2 arctan (x) - \frac{1}{x} + C

Solution steps

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

Expand

\frac{3 x ^{2} + 1}{x ^{2} ( 1 + x ^{2} )} : \frac{3 x ^{2}}{x ^{2} ( 1 + x ^{2} )} + \frac{1}{x ^{2} ( 1 + x ^{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{3 x ^{2}}{x ^{2} ( 1 + x ^{2} )} d x + \int \frac{1}{x ^{2} ( 1 + x ^{2} )} d x

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

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

= 3 arctan (x) - \frac{1}{x} - arctan (x)

Simplify

= 2 arctan (x) - \frac{1}{x}

Add a constant to the solution

= 2 arctan (x) - \frac{1}{x} + C

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