# What is #f(x) = int -cos6x -3tanx+cot(x/3) dx# if #f(pi)=-2 #?

,where c= integration constant

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To find the function ( f(x) ), integrate the given expression:

[ f(x) = \int (-\cos^6(x) - 3\tan(x) + \cot\left(\frac{x}{3}\right)) , dx ]

Given ( f(\pi) = -2 ), we have:

[ -2 = \int (-\cos^6(\pi) - 3\tan(\pi) + \cot\left(\frac{\pi}{3}\right)) , d\pi ]

[ -2 = \int (-1 - 0 + \sqrt{3}) , d\pi ]

[ -2 = \left[-\pi + \sqrt{3}\pi\right] + C ]

[ -2 = (\sqrt{3} - 1)\pi + C ]

[ C = -2 - (\sqrt{3} - 1)\pi ]

Thus, the function ( f(x) ) is:

[ f(x) = \int (-\cos^6(x) - 3\tan(x) + \cot\left(\frac{x}{3}\right)) , dx + (-2 - (\sqrt{3} - 1)\pi) ]

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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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