What is the integral of (5x)/(x^2+4x+4) ?

The integral of (5x)/(x^2+4x+4) is 5(ln|x+2|+2/(x+2))+C

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integral of (5x)/(x^2+4x+4)

\int \frac{5 x}{x ^{2} + 4 x + 4} d x

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

Solution steps

\int \frac{5 x}{x ^{2} + 4 x + 4} d x

Take the constant out:

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

= 5 \cdot \int \frac{x}{x ^{2} + 4 x + 4} d x

Complete the square

x^{2} + 4 x + 4 : (x + 2)^{2}

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

Apply u-substitution

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

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

Expand

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

= 5 \cdot \int \frac{1}{u} - \frac{2}{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

= 5 (\int \frac{1}{u} d u - \int \frac{2}{u ^{2}} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

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

= 5 (ln ∣ u ∣ - (- \frac{2}{u}))

Substitute back

u = x + 2

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

Simplify

= 5 (ln ∣ x + 2 ∣ + \frac{2}{x + 2})

Add a constant to the solution

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

f (x) = \frac{1}{3 - 2 x}

(3, 5), (0, - 1)

\int 2 sin^{2} (\frac{θ}{2}) d θ

\frac{x}{( 1 - x ^{2} )}

x \to 3 lim (- \frac{7}{x - 3})

What is the integral of (5x)/(x^2+4x+4) ?
The integral of (5x)/(x^2+4x+4) is 5(ln|x+2|+2/(x+2))+C