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

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

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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

= \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 ∣

Add a constant to the solution

= 4 + x - 4 ln ∣ 4 + x ∣ + C

e (t + 1)

\frac{\partial}{\partial y} (3 x^{4} e^{y})

\int \frac{24}{( x ^{2} - 9 )} d x

\int \frac{1}{5 - 4 r - r ^{2}} d r

f (x) = \frac{x}{1 + 5 x ^{2}}, at x = 3