What is the integral of (4x-6)/(x^2+4) ?

The integral of (4x-6)/(x^2+4) is 2ln|x^2+4|-3arctan(x/2)+C

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integral of (4x-6)/(x^2+4)

\int \frac{4 x - 6}{x ^{2} + 4} d x

2 ln x^{2} + 4 - 3 arctan (\frac{x}{2}) + C

Solution steps

\int \frac{4 x - 6}{x ^{2} + 4} d x

Expand

\frac{4 x - 6}{x ^{2} + 4} : \frac{4 x}{x ^{2} + 4} - \frac{6}{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{4 x}{x ^{2} + 4} d x - \int \frac{6}{x ^{2} + 4} d x

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

\int \frac{6}{x ^{2} + 4} d x = 3 arctan (\frac{x}{2})

= 2 ln x^{2} + 4 - 3 arctan (\frac{x}{2})

Add a constant to the solution

= 2 ln x^{2} + 4 - 3 arctan (\frac{x}{2}) + C

\int \frac{3}{2} x^{2} d x

\int \frac{1}{x ^{3} - 2 x ^{2} + x} d x

\frac{\partial}{\partial y} (2 z)

\frac{2 x}{x ^{2} + 1}

\int \frac{4 + 6 x - 3 x ^{2}}{x} d x

What is the integral of (4x-6)/(x^2+4) ?
The integral of (4x-6)/(x^2+4) is 2ln|x^2+4|-3arctan(x/2)+C