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

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

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

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

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

Solution steps

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

Take the partial fraction of

\frac{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

\frac{d}{d x} (\frac{x}{2} - \frac{2}{x})

\frac{d}{d x} ((5 + 3 x)^{5})

x \to \infty lim (- 3 x + x)

\frac{d}{d x} (\frac{4}{5} x^{\frac{3}{2}})

\int x^{2} 2 + x d x

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