What is the integral of (6x+7)/(x^2+4x+4) ?

The integral of (6x+7)/(x^2+4x+4) is 6ln|x+2|+5/(x+2)+C

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integral of (6x+7)/(x^2+4x+4)

\int \frac{6 x + 7}{x ^{2} + 4 x + 4} d x

6 ln ∣ x + 2 ∣ + \frac{5}{x + 2} + C

Solution steps

\int \frac{6 x + 7}{x ^{2} + 4 x + 4} d x

Expand

\frac{6 x + 7}{x ^{2} + 4 x + 4} : \frac{6 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{6 x}{x ^{2} + 4 x + 4} d x + \int \frac{7}{x ^{2} + 4 x + 4} d x

\int \frac{6 x}{x ^{2} + 4 x + 4} d x = 6 (ln ∣ x + 2 ∣ + \frac{2}{x + 2})

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

= 6 (ln ∣ x + 2 ∣ + \frac{2}{x + 2}) - \frac{7}{x + 2}

Simplify

6 (ln ∣ x + 2 ∣ + \frac{2}{x + 2}) - \frac{7}{x + 2} : 6 ln ∣ x + 2 ∣ + \frac{5}{x + 2}

= 6 ln ∣ x + 2 ∣ + \frac{5}{x + 2}

Add a constant to the solution

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

\int_{- 3}^{1} x^{2} e^{x^{3} + 27} d x

\frac{( 3 s + 3 )}{s ^{2} - 9}

\int cos^{3} (6 x) d x

x + y = 0, x + y^{2} = 56

x^{n}

What is the integral of (6x+7)/(x^2+4x+4) ?
The integral of (6x+7)/(x^2+4x+4) is 6ln|x+2|+5/(x+2)+C