What is the integral of (5x-2)/((x+3)^2) ?

The integral of (5x-2)/((x+3)^2) is 5ln|x+3|+(17)/(x+3)+C

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integral of (5x-2)/((x+3)^2)

\int \frac{5 x - 2}{( x + 3 ) ^{2}} d x

5 ln ∣ x + 3 ∣ + \frac{17}{x + 3} + C

Solution steps

\int \frac{5 x - 2}{( x + 3 ) ^{2}} d x

Expand

\frac{5 x - 2}{( x + 3 ) ^{2}} : \frac{5 x}{( x + 3 ) ^{2}} - \frac{2}{( 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{5 x}{( x + 3 ) ^{2}} d x - \int \frac{2}{( x + 3 ) ^{2}} d x

\int \frac{5 x}{( x + 3 ) ^{2}} d x = 5 (ln ∣ x + 3 ∣ + \frac{3}{x + 3})

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

= 5 (ln ∣ x + 3 ∣ + \frac{3}{x + 3}) - (- \frac{2}{x + 3})

Simplify

5 (ln ∣ x + 3 ∣ + \frac{3}{x + 3}) - (- \frac{2}{x + 3}) : 5 ln ∣ x + 3 ∣ + \frac{17}{x + 3}

= 5 ln ∣ x + 3 ∣ + \frac{17}{x + 3}

Add a constant to the solution

= 5 ln ∣ x + 3 ∣ + \frac{17}{x + 3} + C

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What is the integral of (5x-2)/((x+3)^2) ?
The integral of (5x-2)/((x+3)^2) is 5ln|x+3|+(17)/(x+3)+C