What is the integral of (6x+1)/((x+2)^2) ?

The integral of (6x+1)/((x+2)^2) is 6ln|x+2|+(11)/(x+2)+C

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integral of (6x+1)/((x+2)^2)

\int \frac{6 x + 1}{( x + 2 ) ^{2}} d x

6 ln ∣ x + 2 ∣ + \frac{11}{x + 2} + C

Solution steps

\int \frac{6 x + 1}{( x + 2 ) ^{2}} d x

Expand

\frac{6 x + 1}{( x + 2 ) ^{2}} : \frac{6 x}{( x + 2 ) ^{2}} + \frac{1}{( 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{6 x}{( x + 2 ) ^{2}} d x + \int \frac{1}{( x + 2 ) ^{2}} d x

\int \frac{6 x}{( x + 2 ) ^{2}} d x = 6 (ln ∣ x + 2 ∣ + \frac{2}{x + 2})

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

= 6 (ln ∣ x + 2 ∣ + \frac{2}{x + 2}) - \frac{1}{x + 2}

Simplify

6 (ln ∣ x + 2 ∣ + \frac{2}{x + 2}) - \frac{1}{x + 2} : 6 ln ∣ x + 2 ∣ + \frac{11}{x + 2}

= 6 ln ∣ x + 2 ∣ + \frac{11}{x + 2}

Add a constant to the solution

= 6 ln ∣ x + 2 ∣ + \frac{11}{x + 2} + C

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What is the integral of (6x+1)/((x+2)^2) ?
The integral of (6x+1)/((x+2)^2) is 6ln|x+2|+(11)/(x+2)+C