What is the integral of (x+5)/(x-3) ?

The integral of (x+5)/(x-3) is 8ln|x-3|+x+C

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integral of (x+5)/(x-3)

\int \frac{x + 5}{x - 3} d x

8 ln ∣ x - 3 ∣ + x + C

Solution steps

\int \frac{x + 5}{x - 3} d x

\frac{x + 5}{x - 3} = \frac{8}{x - 3} + 1

= \int \frac{8}{x - 3} + 1 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{8}{x - 3} d x + \int 1 d x

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

\int 1 d x = x

= 8 ln ∣ x - 3 ∣ + x

Add a constant to the solution

= 8 ln ∣ x - 3 ∣ + x + C

\int (3 x - 7)^{5} d x

\int (5 - 2 x)^{10} d x

\int 4 x 2 x^{2} + 3 d x

\int sin (\frac{3}{2}) x cos (4 x) d x

\int x^{5} + 5 x d x

What is the integral of (x+5)/(x-3) ?
The integral of (x+5)/(x-3) is 8ln|x-3|+x+C