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

The integral of u/(2u^2+2) is 1/4 ln|2u^2+2|+C

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integral of u/(2u^2+2)

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

\frac{1}{4} ln 2 u^{2} + 2 + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{4 v} d v

= \int \frac{1}{4 v} d v

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{4} ln ∣ v ∣

Substitute back

v = 2 u^{2} + 2

= \frac{1}{4} ln 2 u^{2} + 2

Add a constant to the solution

= \frac{1}{4} ln 2 u^{2} + 2 + C

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What is the integral of u/(2u^2+2) ?
The integral of u/(2u^2+2) is 1/4 ln|2u^2+2|+C