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

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

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

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

- ln x^{2} + 1 + C

Solution steps

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

Apply u-substitution

= - 2 \cdot \int \frac{1}{2 u} d u

Take the constant out:

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

= - 2 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 2 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = x^{2} + 1

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

Simplify

- 2 \cdot \frac{1}{2} ln x^{2} + 1 : - ln x^{2} + 1

= - ln x^{2} + 1

Add a constant to the solution

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

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