What is the integral of ((6x+7)/((x+2)^2)) ?

The integral of ((6x+7)/((x+2)^2)) is 6ln|x+2|+5/(x+2)+C

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integral of ((6x+7)/((x+2)^2))

\int (\frac{6 x + 7}{( x + 2 ) ^{2}}) d x

6 ln ∣ x + 2 ∣ + \frac{5}{x + 2} + C

Solution steps

\int \frac{6 x + 7}{( x + 2 ) ^{2}} d x

Apply u-substitution

: \int \frac{6 u - 5}{u ^{2}} d u

= \int \frac{6 u - 5}{u ^{2}} d u

Expand

\frac{6 u - 5}{u ^{2}} : \frac{6}{u} - \frac{5}{u ^{2}}

= \int \frac{6}{u} - \frac{5}{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{6}{u} d u - \int \frac{5}{u ^{2}} d u

\int \frac{6}{u} d u = 6 ln ∣ u ∣

\int \frac{5}{u ^{2}} d u = - \frac{5}{u}

= 6 ln ∣ u ∣ - (- \frac{5}{u})

Substitute back

u = x + 2

= 6 ln ∣ x + 2 ∣ - (- \frac{5}{x + 2})

Simplify

= 6 ln ∣ x + 2 ∣ + \frac{5}{x + 2}

Add a constant to the solution

= 6 ln ∣ x + 2 ∣ + \frac{5}{x + 2} + C

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What is the integral of ((6x+7)/((x+2)^2)) ?
The integral of ((6x+7)/((x+2)^2)) is 6ln|x+2|+5/(x+2)+C