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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{x ^{2}} d x = - \frac{1}{x}

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

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

Add a constant to the solution

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

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