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

The integral of ((4u^2-8))/(2u^2+1) is 2u-5sqrt(2)arctan(sqrt(2)u)+C

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integral of ((4u^2-8))/(2u^2+1)

\int \frac{( 4 u ^{2} - 8 )}{2 u ^{2} + 1} d u

2 u - 52 arctan (2 u) + C

Solution steps

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

Expand

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

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

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

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

Simplify

2 (- arctan (2 u) + 2 u) - 42 arctan (2 u) : 2 u - 52 arctan (2 u)

= 2 u - 52 arctan (2 u)

Add a constant to the solution

= 2 u - 52 arctan (2 u) + C

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

- \frac{24}{x ^{4}}

\int p (p + 3)^{6} d p

x \to 4 lim (x - 4)

x \to 5 lim (25 - x^{2})

What is the integral of ((4u^2-8))/(2u^2+1) ?
The integral of ((4u^2-8))/(2u^2+1) is 2u-5sqrt(2)arctan(sqrt(2)u)+C