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

The integral of (2x)/(x^3+x) is 2arctan(x)+C

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

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

2 arctan (x) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Simplify

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

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

Use the common integral:

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

= 2 arctan (x)

Add a constant to the solution

= 2 arctan (x) + C

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What is the integral of (2x)/(x^3+x) ?
The integral of (2x)/(x^3+x) is 2arctan(x)+C