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

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

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

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

ln 1 + x^{2} + C

Solution steps

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

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

Simplify

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

= ln 1 + x^{2}

Add a constant to the solution

= ln 1 + x^{2} + C

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