What is the integral of x/((x+3)^2) ?

The integral of x/((x+3)^2) is ln|x+3|+3/(x+3)+C

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integral of x/((x+3)^2)

\int \frac{x}{( x + 3 ) ^{2}} d x

ln ∣ x + 3 ∣ + \frac{3}{x + 3} + C

Solution steps

\int \frac{x}{( x + 3 ) ^{2}} d x

Apply u-substitution

: \int \frac{u - 3}{u ^{2}} d u

= \int \frac{u - 3}{u ^{2}} d u

Expand

\frac{u - 3}{u ^{2}} : \frac{1}{u} - \frac{3}{u ^{2}}

= \int \frac{1}{u} - \frac{3}{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{3}{u ^{2}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{3}{u ^{2}} d u = - \frac{3}{u}

= ln ∣ u ∣ - (- \frac{3}{u})

Substitute back

u = x + 3

= ln ∣ x + 3 ∣ - (- \frac{3}{x + 3})

Simplify

= ln ∣ x + 3 ∣ + \frac{3}{x + 3}

Add a constant to the solution

= ln ∣ x + 3 ∣ + \frac{3}{x + 3} + C

cot (x) y^{^{'}} = y

\frac{\partial}{\partial y} (e^{x^{2}} + y^{2})

x \to 3 + lim (\frac{x ^{2}}{x - 3})

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

\int \frac{1}{x - 5} d x

What is the integral of x/((x+3)^2) ?
The integral of x/((x+3)^2) is ln|x+3|+3/(x+3)+C