# How do you evaluate the integral #int sqrt(4+x^2)#?

You can use identity:

So let:

Considering:

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To evaluate the integral ∫√(4 + x^2), you can use trigonometric substitution. Let x = 2tan(θ), then dx = 2sec^2(θ)dθ. Substituting these into the integral gives ∫√(4 + (2tan(θ))^2) * 2sec^2(θ)dθ. Simplify this to get ∫√(4 + 4tan^2(θ)) * 2sec^2(θ)dθ. Using the trigonometric identity sec^2(θ) = 1 + tan^2(θ), you get ∫√(4sec^2(θ)) * 2sec^2(θ)dθ. This simplifies to ∫2(2sec(θ)) * 2sec^2(θ)dθ. Now, simplify further to get ∫4sec^3(θ)dθ. This integral can be evaluated using the formula for the integral of sec^3(θ), which is (1/2)sec(θ)tan(θ) + (1/2)ln|sec(θ) + tan(θ)| + C. Substitute back θ = arctan(x/2) to get the final answer.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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