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

The integral of x/((2+x)) is 2+x-2ln|2+x|+C

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integral of x/((2+x))

integral

\frac{x}{( 2 + x )}

2 + x - 2 ln ∣ 2 + x ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 2}{u} d u

= \int \frac{u - 2}{u} d u

Expand

\frac{u - 2}{u} : 1 - \frac{2}{u}

= \int 1 - \frac{2}{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{2}{u} d u

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= u - 2 ln ∣ u ∣

Substitute back

u = 2 + x

= 2 + x - 2 ln ∣ 2 + x ∣

Add a constant to the solution

= 2 + x - 2 ln ∣ 2 + x ∣ + C

sin (x) sec^{2} (x)

x^{3} - 2 x^{2} - x + 1

\frac{4}{3 x - 2}

\frac{1}{( 1 + cos ^{2} ( x ) )}

\frac{( y + 1 )}{y ^{3}}