What is the integral of (4x^2+8)/(2x^2+1) ?

The integral of (4x^2+8)/(2x^2+1) is 2x+3sqrt(2)arctan(sqrt(2)x)+C

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integral of (4x^2+8)/(2x^2+1)

\int \frac{4 x ^{2} + 8}{2 x ^{2} + 1} d x

2 x + 32 arctan (2 x) + C

Solution steps

\int \frac{4 x ^{2} + 8}{2 x ^{2} + 1} d x

Expand

\frac{4 x ^{2} + 8}{2 x ^{2} + 1} : \frac{4 x ^{2}}{2 x ^{2} + 1} + \frac{8}{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{4 x ^{2}}{2 x ^{2} + 1} d x + \int \frac{8}{2 x ^{2} + 1} d x

\int \frac{4 x ^{2}}{2 x ^{2} + 1} d x = 2 (- arctan (2 x) + 2 x)

\int \frac{8}{2 x ^{2} + 1} d x = 42 arctan (2 x)

= 2 (- arctan (2 x) + 2 x) + 42 arctan (2 x)

Simplify

2 (- arctan (2 x) + 2 x) + 42 arctan (2 x) : 2 x + 32 arctan (2 x)

= 2 x + 32 arctan (2 x)

Add a constant to the solution

= 2 x + 32 arctan (2 x) + C

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What is the integral of (4x^2+8)/(2x^2+1) ?
The integral of (4x^2+8)/(2x^2+1) is 2x+3sqrt(2)arctan(sqrt(2)x)+C