What is the integral of x/((1+x)) ?

The integral of x/((1+x)) is 1+x-ln|1+x|+C

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integral of x/((1+x))

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

1 + x - ln ∣ 1 + x ∣ + C

Solution steps

\int \frac{x}{1 + x} d x

Apply u-substitution

: \int \frac{u - 1}{u} d u

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

Expand

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

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

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= u - ln ∣ u ∣

Substitute back

u = 1 + x

= 1 + x - ln ∣ 1 + x ∣

Add a constant to the solution

= 1 + x - ln ∣ 1 + x ∣ + C

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