What is the integral of x/(x^4+3x^2+2) ?

The integral of x/(x^4+3x^2+2) is 1/2 ln|x^2+1|-1/2 ln|x^2+2|+C

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integral of x/(x^4+3x^2+2)

\int \frac{x}{x ^{4} + 3 x ^{2} + 2} d x

\frac{1}{2} ln x^{2} + 1 - \frac{1}{2} ln x^{2} + 2 + C

Solution steps

\int \frac{x}{x ^{4} + 3 x ^{2} + 2} d x

Take the partial fraction of

\frac{x}{x ^{4} + 3 x ^{2} + 2} : \frac{x}{x ^{2} + 1} - \frac{x}{x ^{2} + 2}

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

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

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

= \frac{1}{2} ln x^{2} + 1 - \frac{1}{2} ln x^{2} + 2

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 1 - \frac{1}{2} ln x^{2} + 2 + C

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What is the integral of x/(x^4+3x^2+2) ?
The integral of x/(x^4+3x^2+2) is 1/2 ln|x^2+1|-1/2 ln|x^2+2|+C