What is the integral of (5x+7)/(x^2+4x+4) ?

The integral of (5x+7)/(x^2+4x+4) is 5ln|x+2|+3/(x+2)+C

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integral of (5x+7)/(x^2+4x+4)

\int \frac{5 x + 7}{x ^{2} + 4 x + 4} d x

5 ln ∣ x + 2 ∣ + \frac{3}{x + 2} + C

Solution steps

\int \frac{5 x + 7}{x ^{2} + 4 x + 4} d x

Expand

\frac{5 x + 7}{x ^{2} + 4 x + 4} : \frac{5 x}{x ^{2} + 4 x + 4} + \frac{7}{x ^{2} + 4 x + 4}

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

\int \frac{5 x}{x ^{2} + 4 x + 4} d x = 5 (ln ∣ x + 2 ∣ + \frac{2}{x + 2})

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

= 5 (ln ∣ x + 2 ∣ + \frac{2}{x + 2}) - \frac{7}{x + 2}

Simplify

5 (ln ∣ x + 2 ∣ + \frac{2}{x + 2}) - \frac{7}{x + 2} : 5 ln ∣ x + 2 ∣ + \frac{3}{x + 2}

= 5 ln ∣ x + 2 ∣ + \frac{3}{x + 2}

Add a constant to the solution

= 5 ln ∣ x + 2 ∣ + \frac{3}{x + 2} + C

\int \frac{e ^{- c o t (z)}}{sin ^{2} ( z )} d z

\frac{d ^{2}}{d x ^{2}} (\frac{x ^{2}}{( 2 + x )})

\frac{d}{d x} (\frac{x ^{2}}{y ^{2}})

\frac{d}{d x} (\frac{sin ( x )}{2 + cos ( x )})

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

What is the integral of (5x+7)/(x^2+4x+4) ?
The integral of (5x+7)/(x^2+4x+4) is 5ln|x+2|+3/(x+2)+C