What is the integral of (2x)/(x^2+6x+9) ?

The integral of (2x)/(x^2+6x+9) is 2(ln|x+3|+3/(x+3))+C

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integral of (2x)/(x^2+6x+9)

\int \frac{2 x}{x ^{2} + 6 x + 9} d x

2 (ln ∣ x + 3 ∣ + \frac{3}{x + 3}) + C

Solution steps

\int \frac{2 x}{x ^{2} + 6 x + 9} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{x}{x ^{2} + 6 x + 9} d x

Complete the square

x^{2} + 6 x + 9 : (x + 3)^{2}

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

Apply u-substitution

: \int \frac{u - 3}{u ^{2}} d u

= 2 \cdot \int \frac{u - 3}{u ^{2}} d u

Expand

\frac{u - 3}{u ^{2}} : \frac{1}{u} - \frac{3}{u ^{2}}

= 2 \cdot \int \frac{1}{u} - \frac{3}{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

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

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{3}{u ^{2}} d u = - \frac{3}{u}

= 2 (ln ∣ u ∣ - (- \frac{3}{u}))

Substitute back

u = x + 3

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

Simplify

= 2 (ln ∣ x + 3 ∣ + \frac{3}{x + 3})

Add a constant to the solution

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

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What is the integral of (2x)/(x^2+6x+9) ?
The integral of (2x)/(x^2+6x+9) is 2(ln|x+3|+3/(x+3))+C