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

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

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

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

x^{2} + ln ∣ x ∣ + C

Solution steps

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

Simplify

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

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

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 2 x d x = x^{2}

\int \frac{1}{x} d x = ln ∣ x ∣

= x^{2} + ln ∣ x ∣

Add a constant to the solution

= x^{2} + ln ∣ x ∣ + C

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