What is the integral of (-5x-12)/(x^2+3) ?

The integral of (-5x-12)/(x^2+3) is -5/2 ln|(x^2)/3+1|-4sqrt(3)arctan(x/(sqrt(3)))+C

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integral of (-5x-12)/(x^2+3)

\int \frac{- 5 x - 12}{x ^{2} + 3} d x

- \frac{5}{2} ln \frac{x ^{2}}{3} + 1 - 43 arctan (\frac{x}{3}) + C

Solution steps

\int \frac{- 5 x - 12}{x ^{2} + 3} d x

Apply Integral Substitution

= \int \frac{- 5 3 u - 12}{3 ( u ^{2} + 1 )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int \frac{- 5 3 u - 12}{u ^{2} + 1} d u

Expand

\frac{- 5 3 u - 12}{u ^{2} + 1} : - \frac{5 3 u}{u ^{2} + 1} - \frac{12}{u ^{2} + 1}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{3} (- \int \frac{5 3 u}{u ^{2} + 1} d u - \int \frac{12}{u ^{2} + 1} d u)

\int \frac{5 3 u}{u ^{2} + 1} d u = \frac{5 3}{2} ln u^{2} + 1

\int \frac{12}{u ^{2} + 1} d u = 12 arctan (u)

= \frac{1}{3} (- \frac{5 3}{2} ln u^{2} + 1 - 12 arctan (u))

Substitute back

u = \frac{x}{3}

= \frac{1}{3} (- \frac{5 3}{2} ln (\frac{x}{3})^{2} + 1 - 12 arctan (\frac{x}{3}))

Simplify

\frac{1}{3} (- \frac{5 3}{2} ln (\frac{x}{3})^{2} + 1 - 12 arctan (\frac{x}{3})) : - \frac{5}{2} ln \frac{x ^{2}}{3} + 1 - 43 arctan (\frac{x}{3})

= - \frac{5}{2} ln \frac{x ^{2}}{3} + 1 - 43 arctan (\frac{x}{3})

Add a constant to the solution

= - \frac{5}{2} ln \frac{x ^{2}}{3} + 1 - 43 arctan (\frac{x}{3}) + C

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What is the integral of (-5x-12)/(x^2+3) ?
The integral of (-5x-12)/(x^2+3) is -5/2 ln|(x^2)/3+1|-4sqrt(3)arctan(x/(sqrt(3)))+C