What is the integral of (x+7)/(x^2+4x+4) ?

The integral of (x+7)/(x^2+4x+4) is ln|x+2|-5/(x+2)+C

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integral of (x+7)/(x^2+4x+4)

\int \frac{x + 7}{x ^{2} + 4 x + 4} d x

ln ∣ x + 2 ∣ - \frac{5}{x + 2} + C

Solution steps

\int \frac{x + 7}{x ^{2} + 4 x + 4} d x

Apply u-substitution

= \int \frac{u}{u ^{2} - 10 u + 25} d u

Complete the square

u^{2} - 10 u + 25 : (u - 5)^{2}

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

Apply u-substitution

= \int \frac{v + 5}{v ^{2}} d v

Expand

\frac{v + 5}{v ^{2}} : \frac{1}{v} + \frac{5}{v ^{2}}

= \int \frac{1}{v} + \frac{5}{v ^{2}} d v

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{1}{v} d v + \int \frac{5}{v ^{2}} d v

\int \frac{1}{v} d v = ln ∣ v ∣

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

= ln ∣ v ∣ - \frac{5}{v}

Substitute back

= ln ∣ x + 7 - 5 ∣ - \frac{5}{x + 7 - 5}

Simplify

= ln ∣ x + 2 ∣ - \frac{5}{x + 2}

Add a constant to the solution

= ln ∣ x + 2 ∣ - \frac{5}{x + 2} + C

What is the integral of (x+7)/(x^2+4x+4) ?
The integral of (x+7)/(x^2+4x+4) is ln|x+2|-5/(x+2)+C