What is the integral of (2x)/(x^2-16)+1/x+2 ?

The integral of (2x)/(x^2-16)+1/x+2 is ln|x^2-16|+ln|x|+2x+C

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integral of (2x)/(x^2-16)+1/x+2

\int \frac{2 x}{x ^{2} - 16} + \frac{1}{x} + 2 d x

ln x^{2} - 16 + ln ∣ x ∣ + 2 x + C

Solution steps

\int \frac{2 x}{x ^{2} - 16} + \frac{1}{x} + 2 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{2 x}{x ^{2} - 16} d x + \int \frac{1}{x} d x + \int 2 d x

\int \frac{2 x}{x ^{2} - 16} d x = ln x^{2} - 16

\int \frac{1}{x} d x = ln ∣ x ∣

\int 2 d x = 2 x

= ln x^{2} - 16 + ln ∣ x ∣ + 2 x

Add a constant to the solution

= ln x^{2} - 16 + ln ∣ x ∣ + 2 x + C

What is the integral of (2x)/(x^2-16)+1/x+2 ?
The integral of (2x)/(x^2-16)+1/x+2 is ln|x^2-16|+ln|x|+2x+C