What is the integral of (2x+4)/(x^2-10x+16) ?

The integral of (2x+4)/(x^2-10x+16) is -4/3 ln|x-2|+10/3 ln|x-8|+C

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integral of (2x+4)/(x^2-10x+16)

\int \frac{2 x + 4}{x ^{2} - 10 x + 16} d x

- \frac{4}{3} ln ∣ x - 2 ∣ + \frac{10}{3} ln ∣ x - 8 ∣ + C

Solution steps

\int \frac{2 x + 4}{x ^{2} - 10 x + 16} d x

Take the partial fraction of

\frac{2 x + 4}{x ^{2} - 10 x + 16} : - \frac{4}{3 ( x - 2 )} + \frac{10}{3 ( x - 8 )}

= \int - \frac{4}{3 ( x - 2 )} + \frac{10}{3 ( x - 8 )} d x

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{4}{3 ( x - 2 )} d x + \int \frac{10}{3 ( x - 8 )} d x

\int \frac{4}{3 ( x - 2 )} d x = \frac{4}{3} ln ∣ x - 2 ∣

\int \frac{10}{3 ( x - 8 )} d x = \frac{10}{3} ln ∣ x - 8 ∣

= - \frac{4}{3} ln ∣ x - 2 ∣ + \frac{10}{3} ln ∣ x - 8 ∣

Add a constant to the solution

= - \frac{4}{3} ln ∣ x - 2 ∣ + \frac{10}{3} ln ∣ x - 8 ∣ + C

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What is the integral of (2x+4)/(x^2-10x+16) ?
The integral of (2x+4)/(x^2-10x+16) is -4/3 ln|x-2|+10/3 ln|x-8|+C