What is the integral of (x^2+4x+2)/(x^2+1) ?

The integral of (x^2+4x+2)/(x^2+1) is x+arctan(x)+2ln|x^2+1|+C

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integral of (x^2+4x+2)/(x^2+1)

\int \frac{x ^{2} + 4 x + 2}{x ^{2} + 1} d x

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

Solution steps

\int \frac{x ^{2} + 4 x + 2}{x ^{2} + 1} d x

Expand

\frac{x ^{2} + 4 x + 2}{x ^{2} + 1} : \frac{x ^{2}}{x ^{2} + 1} + \frac{4 x}{x ^{2} + 1} + \frac{2}{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{x ^{2}}{x ^{2} + 1} d x + \int \frac{4 x}{x ^{2} + 1} d x + \int \frac{2}{x ^{2} + 1} d x

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

\int \frac{4 x}{x ^{2} + 1} d x = 2 ln x^{2} + 1

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (x^2+4x+2)/(x^2+1) ?
The integral of (x^2+4x+2)/(x^2+1) is x+arctan(x)+2ln|x^2+1|+C