# How do you evaluate #int 1/(1+4x^2)# from #[0,sqrt3/2]#?

The answer is

We evaluate this integral by substitution

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You integrate by reconducing to the fundamental integral with arcotangent primitive

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To evaluate ( \int_0^{\frac{\sqrt{3}}{2}} \frac{1}{{1 + 4x^2}} , dx ), use the trigonometric substitution method. Let ( x = \frac{\sqrt{3}}{2} \tan(\theta) ), then ( dx = \frac{\sqrt{3}}{2} \sec^2(\theta) , d\theta ). Substitute these into the integral and integrate with respect to ( \theta ) from ( 0 ) to ( \frac{\pi}{3} ). After integration, substitute back ( x = \frac{\sqrt{3}}{2} \tan(\theta) ) to get the final result.

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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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