What is the integral of (4x+6)/(5(x^2+9)) ?

The integral of (4x+6)/(5(x^2+9)) is 1/5 (2ln|x^2+9|+2arctan(x/3))+C

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integral of (4x+6)/(5(x^2+9))

\int \frac{4 x + 6}{5 ( x ^{2} + 9 )} d x

\frac{1}{5} (2 ln x^{2} + 9 + 2 arctan (\frac{x}{3})) + C

Solution steps

\int \frac{4 x + 6}{5 ( x ^{2} + 9 )} d x

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{4 x + 6}{x ^{2} + 9} d x

Expand

\frac{4 x + 6}{x ^{2} + 9} : \frac{4 x}{x ^{2} + 9} + \frac{6}{x ^{2} + 9}

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}{5} (\int \frac{4 x}{x ^{2} + 9} d x + \int \frac{6}{x ^{2} + 9} d x)

\int \frac{4 x}{x ^{2} + 9} d x = 2 ln x^{2} + 9

\int \frac{6}{x ^{2} + 9} d x = 2 arctan (\frac{x}{3})

= \frac{1}{5} (2 ln x^{2} + 9 + 2 arctan (\frac{x}{3}))

Add a constant to the solution

= \frac{1}{5} (2 ln x^{2} + 9 + 2 arctan (\frac{x}{3})) + C

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What is the integral of (4x+6)/(5(x^2+9)) ?
The integral of (4x+6)/(5(x^2+9)) is 1/5 (2ln|x^2+9|+2arctan(x/3))+C