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

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

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integral of x/((a+x)^2)

\int \frac{x}{( a + x ) ^{2}} d x

ln ∣ a + x ∣ + \frac{a}{a + x} + C

Solution steps

\int \frac{x}{( a + x ) ^{2}} d x

Apply u-substitution

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

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

Expand

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

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

= ln ∣ u ∣ - (- \frac{a}{u})

Substitute back

u = a + x

= ln ∣ a + x ∣ - (- \frac{a}{a + x})

Simplify

= ln ∣ a + x ∣ + \frac{a}{a + x}

Add a constant to the solution

= ln ∣ a + x ∣ + \frac{a}{a + x} + C

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What is the integral of x/((a+x)^2) ?
The integral of x/((a+x)^2) is ln|a+x|+a/(a+x)+C