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

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

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integral of u/(1+u^2)

\int \frac{u}{1 + u ^{2}} d u

\frac{1}{2} ln 1 + u^{2} + C

Solution steps

\int \frac{u}{1 + u ^{2}} d u

Apply u-substitution

= \int \frac{1}{2 v} d v

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2} ln ∣ v ∣

Substitute back

v = 1 + u^{2}

= \frac{1}{2} ln 1 + u^{2}

Add a constant to the solution

= \frac{1}{2} ln 1 + u^{2} + C

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