What is the integral of ((6x+3))/((x^2+1)) ?

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

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integral of ((6x+3))/((x^2+1))

integral

\frac{( 6 x + 3 )}{( x ^{2} + 1 )}

3 ln x^{2} + 1 + 3 arctan (x) + C

Solution steps

\int \frac{6 x + 3}{x ^{2} + 1} d x

Expand

\frac{6 x + 3}{x ^{2} + 1} : \frac{6 x}{x ^{2} + 1} + \frac{3}{x ^{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{6 x}{x ^{2} + 1} d x + \int \frac{3}{x ^{2} + 1} d x

\int \frac{6 x}{x ^{2} + 1} d x = 3 ln x^{2} + 1

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

= 3 ln x^{2} + 1 + 3 arctan (x)

Add a constant to the solution

= 3 ln x^{2} + 1 + 3 arctan (x) + C

\frac{( 1 + a )}{( 1 - a )}

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

\frac{1}{cos ^{2} ( 1 - 2 x )}

\frac{1}{2 x ^{2} y}

x^{4} e^{x}

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