What is the integral of (u+2)/(5u+6) ?

The integral of (u+2)/(5u+6) is 1/25 (5u+6+4ln|5u+6|)+C

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integral of (u+2)/(5u+6)

\int \frac{u + 2}{5 u + 6} d u

\frac{1}{25} (5 u + 6 + 4 ln ∣ 5 u + 6 ∣) + C

Solution steps

\int \frac{u + 2}{5 u + 6} d u

Apply u-substitution

: \int \frac{v + 4}{25 v} d v

= \int \frac{v + 4}{25 v} d v

Take the constant out:

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

= \frac{1}{25} \cdot \int \frac{v + 4}{v} d v

Expand

\frac{v + 4}{v} : 1 + \frac{4}{v}

= \frac{1}{25} \cdot \int 1 + \frac{4}{v} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{25} (\int 1 d v + \int \frac{4}{v} d v)

\int 1 d v = v

\int \frac{4}{v} d v = 4 ln ∣ v ∣

= \frac{1}{25} (v + 4 ln ∣ v ∣)

Substitute back

v = 5 u + 6

= \frac{1}{25} (5 u + 6 + 4 ln ∣ 5 u + 6 ∣)

Add a constant to the solution

= \frac{1}{25} (5 u + 6 + 4 ln ∣ 5 u + 6 ∣) + C

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What is the integral of (u+2)/(5u+6) ?
The integral of (u+2)/(5u+6) is 1/25 (5u+6+4ln|5u+6|)+C