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

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

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

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

ln x^{2} + 1 + 6 arctan (x) + C

Solution steps

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

Expand

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

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

\int \frac{6}{x ^{2} + 1} d x = 6 arctan (x)

= ln x^{2} + 1 + 6 arctan (x)

Add a constant to the solution

= ln x^{2} + 1 + 6 arctan (x) + C

x \to 2 - lim (3 - x)

\frac{d}{d x} (csc (4 x^{4} - 2))

x \to 0 lim ((x^{2}) cos (\frac{7}{x}))

y = 9 x^{2}, y^{2} = \frac{1}{9} x

\int x^{2} e^{12 x} d x

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