What is the integral of u/((2u+11)) ?

The integral of u/((2u+11)) is 1/4 (2u+11-11ln|2u+11|)+C

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integral of u/((2u+11))

\int \frac{u}{( 2 u + 11 )} d u

\frac{1}{4} (2 u + 11 - 11 ln ∣ 2 u + 11 ∣) + C

Solution steps

\int \frac{u}{2 u + 11} d u

Apply u-substitution

: \int \frac{v - 11}{4 v} d v

= \int \frac{v - 11}{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{v - 11}{v} d v

Expand

\frac{v - 11}{v} : 1 - \frac{11}{v}

= \frac{1}{4} \cdot \int 1 - \frac{11}{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}{4} (\int 1 d v - \int \frac{11}{v} d v)

\int 1 d v = v

\int \frac{11}{v} d v = 11 ln ∣ v ∣

= \frac{1}{4} (v - 11 ln ∣ v ∣)

Substitute back

v = 2 u + 11

= \frac{1}{4} (2 u + 11 - 11 ln ∣ 2 u + 11 ∣)

Add a constant to the solution

= \frac{1}{4} (2 u + 11 - 11 ln ∣ 2 u + 11 ∣) + C

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What is the integral of u/((2u+11)) ?
The integral of u/((2u+11)) is 1/4 (2u+11-11ln|2u+11|)+C