# Why can't you integrate #sqrt(1+(cosx/-sinx)^2#?

First of all it's useful to "change" a little the function:

Before integrate this function, it is useful to remember the parametric formula of sinus, that says:

Now the integral will be done with the method of substitution:

Our integral becoms:

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The integral of ( \sqrt{1 + \left(\frac{\cos(x)}{-\sin(x)}\right)^2} ) cannot be integrated in terms of elementary functions because it does not have an antiderivative that can be expressed using standard functions like polynomials, exponentials, trigonometric functions, and their inverses. This integral involves a trigonometric expression within the square root, which leads to a complex integrand. While certain techniques like trigonometric substitution or integration by parts may be attempted, they often result in expressions that cannot be simplified further. As a result, the integral remains in a form that cannot be evaluated using standard methods, making it non-elementary.

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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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- How do you use the second fundamental theorem of Calculus to find the derivative of given #int (cos(t^3))# from #[cosx, 7x]#?

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