What is the integral of ((16-x))/((4+x)) ?

The integral of ((16-x))/((4+x)) is -x+20ln|x+4|-4+C

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integral of ((16-x))/((4+x))

integral

\frac{( 16 - x )}{( 4 + x )}

- x + 20 ln ∣ x + 4 ∣ - 4 + C

Solution steps

\int \frac{16 - x}{4 + x} d x

Expand

\frac{16 - x}{4 + x} : \frac{16}{4 + x} - \frac{x}{4 + x}

Apply the Sum Rule:

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

= \int \frac{16}{4 + x} d x - \int \frac{x}{4 + x} d x

\int \frac{16}{4 + x} d x = 16 ln ∣ 4 + x ∣

\int \frac{x}{4 + x} d x = 4 + x - 4 ln ∣ 4 + x ∣

= 16 ln ∣ 4 + x ∣ - (4 + x - 4 ln ∣ 4 + x ∣)

Simplify

= - x + 20 ln ∣ x + 4 ∣ - 4

Add a constant to the solution

= - x + 20 ln ∣ x + 4 ∣ - 4 + C

What is the integral of ((16-x))/((4+x)) ?
The integral of ((16-x))/((4+x)) is -x+20ln|x+4|-4+C