What is the integral of (3x+4)/((x^2+9)) ?

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

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integral of (3x+4)/((x^2+9))

\int \frac{3 x + 4}{( x ^{2} + 9 )} d x

\frac{3}{2} ln x^{2} + 9 + \frac{4}{3} arctan (\frac{x}{3}) + C

Solution steps

\int \frac{3 x + 4}{x ^{2} + 9} d x

Expand

\frac{3 x + 4}{x ^{2} + 9} : \frac{3 x}{x ^{2} + 9} + \frac{4}{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

= \int \frac{3 x}{x ^{2} + 9} d x + \int \frac{4}{x ^{2} + 9} d x

\int \frac{3 x}{x ^{2} + 9} d x = \frac{3}{2} ln x^{2} + 9

\int \frac{4}{x ^{2} + 9} d x = \frac{4}{3} arctan (\frac{x}{3})

= \frac{3}{2} ln x^{2} + 9 + \frac{4}{3} arctan (\frac{x}{3})

Add a constant to the solution

= \frac{3}{2} ln x^{2} + 9 + \frac{4}{3} arctan (\frac{x}{3}) + C

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