What is the integral of (8+3x)/(x(x^2+4)) ?

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

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integral of (8+3x)/(x(x^2+4))

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

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

Solution steps

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

Expand

\frac{8 + 3 x}{x ( x ^{2} + 4 )} : \frac{8}{x ( x ^{2} + 4 )} + \frac{3 x}{x ( 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{8}{x ( x ^{2} + 4 )} d x + \int \frac{3 x}{x ( x ^{2} + 4 )} d x

\int \frac{8}{x ( x ^{2} + 4 )} d x = 2 ln ∣ x ∣ - ln x^{2} + 4

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

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

Add a constant to the solution

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

x^{3} - 15 x^{2} + 72 x

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

\int \frac{1}{12 x - x ^{2}} d x

\frac{d}{d x} (\frac{ln ( 5 x )}{5 x})

1. 5^{x}

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