What is the integral of ((x+1))/((x+3)^2) ?

The integral of ((x+1))/((x+3)^2) is ln|x+3|+2/(x+3)+C

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integral of ((x+1))/((x+3)^2)

\int \frac{( x + 1 )}{( x + 3 ) ^{2}} d x

ln ∣ x + 3 ∣ + \frac{2}{x + 3} + C

Solution steps

\int \frac{x + 1}{( x + 3 ) ^{2}} d x

Expand

\frac{x + 1}{( x + 3 ) ^{2}} : \frac{x}{( x + 3 ) ^{2}} + \frac{1}{( x + 3 ) ^{2}}

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{x}{( x + 3 ) ^{2}} d x + \int \frac{1}{( x + 3 ) ^{2}} d x

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

\int \frac{1}{( x + 3 ) ^{2}} d x = - \frac{1}{x + 3}

= ln ∣ x + 3 ∣ + \frac{3}{x + 3} - \frac{1}{x + 3}

Simplify

ln ∣ x + 3 ∣ + \frac{3}{x + 3} - \frac{1}{x + 3} : ln ∣ x + 3 ∣ + \frac{2}{x + 3}

= ln ∣ x + 3 ∣ + \frac{2}{x + 3}

Add a constant to the solution

= ln ∣ x + 3 ∣ + \frac{2}{x + 3} + C

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What is the integral of ((x+1))/((x+3)^2) ?
The integral of ((x+1))/((x+3)^2) is ln|x+3|+2/(x+3)+C