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

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

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integral of 1/(x+x^3)

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

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

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1

Add a constant to the solution

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

\frac{\partial}{\partial x} (e^{- \frac{x ^{2}}{6} - \frac{y ^{2}}{6}})

\int (\frac{5 x ^{4}}{( 10 + x ^{5} ) ^{4}}) d x

n = 1 \sum \infty (cos (1.6))^{n}

\int arcsin (2 x) d x

\frac{\partial}{\partial y} (e^{y} + e^{2 x y})

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