What is the integral of u/(u^2+4) ?

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

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integral of u/(u^2+4)

\int \frac{u}{u ^{2} + 4} d u

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

Solution steps

\int \frac{u}{u ^{2} + 4} d u

Apply u-substitution

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

= \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 = u^{2} + 4

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

Add a constant to the solution

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

y = 6 x

y = cos (tan (x))

\int_{0}^{π} 3 sin^{3} (x) d x

\frac{d}{d x} ((ln (x))^{2} cos (x))

\frac{d}{d t} (e^{t^{2}})

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