What is the integral of ((2x+2))/(x^2+10x+24) ?

The integral of ((2x+2))/(x^2+10x+24) is ln|x^2+10x+24|+4ln|x+6|-4ln|x+4|+C

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integral of ((2x+2))/(x^2+10x+24)

\int \frac{( 2 x + 2 )}{x ^{2} + 10 x + 24} d x

ln x^{2} + 10 x + 24 + 4 ln ∣ x + 6 ∣ - 4 ln ∣ x + 4 ∣ + C

Solution steps

\int \frac{2 x + 2}{x ^{2} + 10 x + 24} d x

Expand

\frac{2 x + 2}{x ^{2} + 10 x + 24} : \frac{2 x}{x ^{2} + 10 x + 24} + \frac{2}{x ^{2} + 10 x + 24}

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{2 x}{x ^{2} + 10 x + 24} d x + \int \frac{2}{x ^{2} + 10 x + 24} d x

\int \frac{2 x}{x ^{2} + 10 x + 24} d x = ln x^{2} + 10 x + 24 + 5 ln ∣ x + 6 ∣ - 5 ln ∣ x + 4 ∣

\int \frac{2}{x ^{2} + 10 x + 24} d x = ln ∣ x + 4 ∣ - ln ∣ x + 6 ∣

= ln x^{2} + 10 x + 24 + 5 ln ∣ x + 6 ∣ - 5 ln ∣ x + 4 ∣ + ln ∣ x + 4 ∣ - ln ∣ x + 6 ∣

Simplify

= ln x^{2} + 10 x + 24 + 4 ln ∣ x + 6 ∣ - 4 ln ∣ x + 4 ∣

Add a constant to the solution

= ln x^{2} + 10 x + 24 + 4 ln ∣ x + 6 ∣ - 4 ln ∣ x + 4 ∣ + C

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What is the integral of ((2x+2))/(x^2+10x+24) ?
The integral of ((2x+2))/(x^2+10x+24) is ln|x^2+10x+24|+4ln|x+6|-4ln|x+4|+C