What is the integral of ((12x+2))/(x^2+4) ?

The integral of ((12x+2))/(x^2+4) is 6ln|x^2+4|+arctan(x/2)+C

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integral of ((12x+2))/(x^2+4)

\int \frac{( 12 x + 2 )}{x ^{2} + 4} d x

6 ln x^{2} + 4 + arctan (\frac{x}{2}) + C

Solution steps

\int \frac{12 x + 2}{x ^{2} + 4} d x

Expand

\frac{12 x + 2}{x ^{2} + 4} : \frac{12 x}{x ^{2} + 4} + \frac{2}{x ^{2} + 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{12 x}{x ^{2} + 4} d x + \int \frac{2}{x ^{2} + 4} d x

\int \frac{12 x}{x ^{2} + 4} d x = 6 ln x^{2} + 4

\int \frac{2}{x ^{2} + 4} d x = arctan (\frac{x}{2})

= 6 ln x^{2} + 4 + arctan (\frac{x}{2})

Add a constant to the solution

= 6 ln x^{2} + 4 + arctan (\frac{x}{2}) + C

What is the integral of ((12x+2))/(x^2+4) ?
The integral of ((12x+2))/(x^2+4) is 6ln|x^2+4|+arctan(x/2)+C