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

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

integral of x/(4-x)

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

4 - x - 4 ln ∣ 4 - x ∣ + C

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

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

Expand

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

= - \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

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

\int 1 d u = u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

= - (- u + 4 ln ∣ u ∣)

Substitute back

u = 4 - x

= - (- (4 - x) + 4 ln ∣ 4 - x ∣)

Simplify

= 4 - x - 4 ln ∣ 4 - x ∣

Add a constant to the solution

= 4 - x - 4 ln ∣ 4 - x ∣ + C

d er i v a t i v e e^{(e^{x})}

d er i v a t i v e y = (\frac{x ^{2} + 4}{x ^{2} - 4})^{3}

x \to 6 - lim (\frac{x}{x - 6})

\int \frac{1}{( x + 2 ) ( x + 2 )} d x

d er i v a t i v e f (x) = sin^{5} (x)