What is the integral of (2x+5)/((x+2)^2) ?

The integral of (2x+5)/((x+2)^2) is 2ln|x+2|-1/(x+2)+C

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integral of (2x+5)/((x+2)^2)

\int \frac{2 x + 5}{( x + 2 ) ^{2}} d x

2 ln ∣ x + 2 ∣ - \frac{1}{x + 2} + C

Solution steps

\int \frac{2 x + 5}{( x + 2 ) ^{2}} d x

Expand

\frac{2 x + 5}{( x + 2 ) ^{2}} : \frac{2 x}{( x + 2 ) ^{2}} + \frac{5}{( x + 2 ) ^{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{2 x}{( x + 2 ) ^{2}} d x + \int \frac{5}{( x + 2 ) ^{2}} d x

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

\int \frac{5}{( x + 2 ) ^{2}} d x = - \frac{5}{x + 2}

= 2 (ln ∣ x + 2 ∣ + \frac{2}{x + 2}) - \frac{5}{x + 2}

Simplify

2 (ln ∣ x + 2 ∣ + \frac{2}{x + 2}) - \frac{5}{x + 2} : 2 ln ∣ x + 2 ∣ - \frac{1}{x + 2}

= 2 ln ∣ x + 2 ∣ - \frac{1}{x + 2}

Add a constant to the solution

= 2 ln ∣ x + 2 ∣ - \frac{1}{x + 2} + C

What is the integral of (2x+5)/((x+2)^2) ?
The integral of (2x+5)/((x+2)^2) is 2ln|x+2|-1/(x+2)+C