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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{2 ( x + 1 )} d x = \frac{1}{2} ln ∣ x + 1 ∣

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

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

Add a constant to the solution

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

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