What is the integral of (2x+3)/(4x+5) ?

The integral of (2x+3)/(4x+5) is 1/8 (4x+5+ln|4x+5|)+C

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integral of (2x+3)/(4x+5)

\int \frac{2 x + 3}{4 x + 5} d x

\frac{1}{8} (4 x + 5 + ln ∣ 4 x + 5 ∣) + C

Solution steps

\int \frac{2 x + 3}{4 x + 5} d x

Apply u-substitution

: \int \frac{u + 1}{8 u} d u

= \int \frac{u + 1}{8 u} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int \frac{u + 1}{u} d u

Expand

\frac{u + 1}{u} : 1 + \frac{1}{u}

= \frac{1}{8} \cdot \int 1 + \frac{1}{u} d u

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}{8} (\int 1 d u + \int \frac{1}{u} d u)

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{1}{8} (u + ln ∣ u ∣)

Substitute back

u = 4 x + 5

= \frac{1}{8} (4 x + 5 + ln ∣ 4 x + 5 ∣)

Add a constant to the solution

= \frac{1}{8} (4 x + 5 + ln ∣ 4 x + 5 ∣) + C

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What is the integral of (2x+3)/(4x+5) ?
The integral of (2x+3)/(4x+5) is 1/8 (4x+5+ln|4x+5|)+C