# How do you evaluate the integral #int 1/(x^2sqrt(4x^2+1))#?

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To evaluate the integral ( \int \frac{1}{x^2\sqrt{4x^2 + 1}} , dx ), you can use trigonometric substitution. Let ( x = \frac{1}{2}\tan(\theta) ). Then, ( dx = \frac{1}{2}\sec^2(\theta) , d\theta ), and ( \sqrt{4x^2 + 1} = \sqrt{\tan^2(\theta) + 1} = \sec(\theta) ).

Substitute these expressions into the integral:

[ \int \frac{1}{x^2\sqrt{4x^2 + 1}} , dx = \int \frac{1}{\left(\frac{1}{2}\tan(\theta)\right)^2 \cdot \sec(\theta)} \cdot \frac{1}{2}\sec^2(\theta) , d\theta ]

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

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

This integral can be evaluated using a trigonometric identity. Recall that ( \sec^2(\theta) = 1 + \tan^2(\theta) ). Therefore, ( \sin^2(\theta) = 1 - \cos^2(\theta) ).

So, ( \int \frac{1}{\sin^2(\theta)} , d\theta = \int \frac{1}{1 - \cos^2(\theta)} , d\theta ).

Using the identity ( \cos(2\theta) = 1 - 2\sin^2(\theta) ), we have ( \sin^2(\theta) = \frac{1}{2}(1 - \cos(2\theta)) ).

Substitute ( u = 2\theta ), then ( du = 2d\theta ), leading to:

[ \int \frac{1}{1 - \cos^2(\theta)} , d\theta = \frac{1}{2} \int \frac{1}{1 - \frac{1}{2}(1 - \cos(u))} , du ]

[ = \frac{1}{2} \int \frac{1}{\frac{1}{2}\cos(u) + \frac{1}{2}} , du ]

[ = \int \frac{1}{\cos(u) + 1} , du ]

[ = \int \frac{1}{2\cos^2\left(\frac{u}{2}\right)} , du ]

This integral can be solved using the standard integral formula for ( \sec^2(u) ).

After integrating, remember to back-substitute to express the result in terms of ( x ).

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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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