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

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

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

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

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

Solution steps

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

Apply Integral Substitution

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

Expand

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

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

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

\int \frac{5}{u ^{2} + 1} d u = 5 arctan (u)

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

Substitute back

u = \frac{x}{5}

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

(\frac{x}{5})^{2} = \frac{x ^{2}}{5}

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

Add a constant to the solution

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

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