# How do you integrate #int 1/sqrt(1-4x^2)# by trigonometric substitution?

Then:

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To integrate ( \int \frac{1}{\sqrt{1-4x^2}} ) using trigonometric substitution, perform the following steps:

- Let ( x = \frac{1}{2}\sin(\theta) ), then ( dx = \frac{1}{2}\cos(\theta) , d\theta ).
- Substitute ( x = \frac{1}{2}\sin(\theta) ) and ( dx = \frac{1}{2}\cos(\theta) , d\theta ) into the integral.
- Simplify the expression under the square root.
- Integrate the simplified expression with respect to ( \theta ).
- Substitute back the original variable ( x ) using trigonometric identities.

This procedure yields the integral in terms of ( \theta ), which can then be evaluated using trigonometric techniques.

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To integrate (\int \frac{1}{\sqrt{1-4x^2}}) using trigonometric substitution, we can make the substitution (x = \frac{1}{2} \sin(\theta)). Then, (dx = \frac{1}{2} \cos(\theta) d\theta) and (\sqrt{1 - 4x^2} = \sqrt{1 - 4\left(\frac{1}{2} \sin(\theta)\right)^2} = \sqrt{1 - \sin^2(\theta)} = \cos(\theta)).

So, after substitution, the integral becomes:

[ \int \frac{1}{\sqrt{1-4x^2}} , dx = \int \frac{1}{\sqrt{1 - 4\left(\frac{1}{2} \sin(\theta)\right)^2}} \cdot \frac{1}{2} \cos(\theta) , d\theta ]

[ = \int \frac{1}{\sqrt{1 - \sin^2(\theta)}} \cdot \frac{1}{2} \cos(\theta) , d\theta ]

[ = \int \frac{1}{\cos(\theta)} \cdot \frac{1}{2} \cos(\theta) , d\theta ]

[ = \frac{1}{2} \int d\theta ]

[ = \frac{\theta}{2} + C ]

Now, we need to revert back to the original variable, (x). Recall the substitution (x = \frac{1}{2} \sin(\theta)). Thus, (\theta = \sin^{-1}(2x)).

Substituting this back into the result, we have:

[ \int \frac{1}{\sqrt{1-4x^2}} , dx = \frac{\sin^{-1}(2x)}{2} + C ]

Where (C) is the constant of integration.

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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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