What is the integral of 1/(x^5+x^3) ?

The integral of 1/(x^5+x^3) is -ln|x|-1/(2x^2)+1/2 ln|x^2+1|+C

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integral of 1/(x^5+x^3)

\int \frac{1}{x ^{5} + x ^{3}} d x

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

Solution steps

\int \frac{1}{x ^{5} + x ^{3}} d x

Take the partial fraction of

\frac{1}{x ^{5} + x ^{3}} : - \frac{1}{x} + \frac{1}{x ^{3}} + \frac{x}{x ^{2} + 1}

= \int - \frac{1}{x} + \frac{1}{x ^{3}} + \frac{x}{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} d x + \int \frac{1}{x ^{3}} d x + \int \frac{x}{x ^{2} + 1} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{1}{x ^{3}} d x = - \frac{1}{2 x ^{2}}

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

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

Add a constant to the solution

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

What is the integral of 1/(x^5+x^3) ?
The integral of 1/(x^5+x^3) is -ln|x|-1/(2x^2)+1/2 ln|x^2+1|+C