What is the integral of (x+3)/(4x+4) ?

The integral of (x+3)/(4x+4) is 1/4 (x+1-ln(4|x+1|))+3/4 ln|4x+4|+C

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integral of (x+3)/(4x+4)

\int \frac{x + 3}{4 x + 4} d x

\frac{1}{4} (x + 1 - ln (4 ∣ x + 1 ∣)) + \frac{3}{4} ln ∣ 4 x + 4 ∣ + C

Solution steps

\int \frac{x + 3}{4 x + 4} d x

Expand

\frac{x + 3}{4 x + 4} : \frac{x}{4 x + 4} + \frac{3}{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{x}{4 x + 4} d x + \int \frac{3}{4 x + 4} d x

\int \frac{x}{4 x + 4} d x = \frac{1}{4} (x + 1 - ln (4 ∣ x + 1 ∣))

\int \frac{3}{4 x + 4} d x = \frac{3}{4} ln ∣ 4 x + 4 ∣

= \frac{1}{4} (x + 1 - ln (4 ∣ x + 1 ∣)) + \frac{3}{4} ln ∣ 4 x + 4 ∣

Add a constant to the solution

= \frac{1}{4} (x + 1 - ln (4 ∣ x + 1 ∣)) + \frac{3}{4} ln ∣ 4 x + 4 ∣ + C

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What is the integral of (x+3)/(4x+4) ?
The integral of (x+3)/(4x+4) is 1/4 (x+1-ln(4|x+1|))+3/4 ln|4x+4|+C