What is the integral of ((3x-1))/(x^2+9) ?

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

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integral of ((3x-1))/(x^2+9)

\int \frac{( 3 x - 1 )}{x ^{2} + 9} d x

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

Solution steps

\int \frac{3 x - 1}{x ^{2} + 9} d x

Expand

\frac{3 x - 1}{x ^{2} + 9} : \frac{3 x}{x ^{2} + 9} - \frac{1}{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{1}{x ^{2} + 9} d x

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

\int \frac{1}{x ^{2} + 9} d x = \frac{1}{3} arctan (\frac{x}{3})

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

Add a constant to the solution

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

\int 3^{x} e^{x} d x

\frac{d}{d x} (sin (x) + csc (x))

\frac{x d y}{d x} = 2 x e^{x} - y + 6 x^{2}

y^{^{''}} + 2 y^{^{'}} + y = x^{2}

\frac{1}{3} x^{3} - \frac{1}{2} x^{2} - 30 x + 2

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