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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

y = ln (cos (x))

\frac{d}{d x} ((ln (x^{2} - 3))^{3})

\frac{d y}{d x}, x^{2} + y^{2} = 4

y = lo g_{3} (8 x - 1)

\frac{d ^{2}}{d x ^{2}} (1 + x^{2})

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