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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Expand

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

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

\int 1 d u = u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

= \frac{1}{25} (u + 4 ln ∣ u ∣)

Substitute back

u = 5 x + 6

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

Add a constant to the solution

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

\frac{d}{d x} (e^{x} cos (x) - sin (x) e^{x})

y^{^{''}} + 36 y^{^{'}} = 0

x \to \frac{π}{2} + lim (sec (x))

\int x (9 - x^{2}) d x

f (x) = \frac{5 x}{x + 2}, (3, 3)

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