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

The integral of (x^2+2)/(2x^2+1) is -1/(2sqrt(2))arctan(sqrt(2)x)+1/2 x+sqrt(2)arctan(sqrt(2)x)+C

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

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

- \frac{1}{2 2} arctan (2 x) + \frac{1}{2} x + 2 arctan (2 x) + C

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

= - \frac{1}{2 2} arctan (2 x) + \frac{1}{2} x + 2 arctan (2 x) + C

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What is the integral of (x^2+2)/(2x^2+1) ?
The integral of (x^2+2)/(2x^2+1) is -1/(2sqrt(2))arctan(sqrt(2)x)+1/2 x+sqrt(2)arctan(sqrt(2)x)+C