What is #f(x) = int cot2x dx# if #f(pi/8) = 0 #?
we have a log integration.
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To find ( f(x) = \int \cot^2(x) , dx ) given that ( f(\frac{\pi}{8}) = 0 ), we first need to find the antiderivative of ( \cot^2(x) ).
The antiderivative of ( \cot^2(x) ) is ( -\cot(x) ).
Using the given information, we have:
[ f(\frac{\pi}{8}) = \int_{0}^{\frac{\pi}{8}} \cot^2(x) , dx = 0 ]
Now, we can set up the integral:
[ \int_{0}^{\frac{\pi}{8}} \cot^2(x) , dx = -\cot\left(\frac{\pi}{8}\right) - (-\cot(0)) = -\cot\left(\frac{\pi}{8}\right) ]
[ -\cot\left(\frac{\pi}{8}\right) = 0 ]
Thus, ( \cot\left(\frac{\pi}{8}\right) = 0 ).
Since ( \cot\left(\frac{\pi}{8}\right) = \frac{\cos\left(\frac{\pi}{8}\right)}{\sin\left(\frac{\pi}{8}\right)} = 0 ), the numerator ( \cos\left(\frac{\pi}{8}\right) = 0 ).
Therefore, ( f(x) = \int \cot^2(x) , dx = -\cot(x) + 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.
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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