Find the value of the integral of $ cot(x) $ from $ 0 $ to $ frac{pi}{4} $ using the unit circle.

To find the value of the integral of $ \cot(x) $ from $ 0 $ to $ \frac{\pi}{4} $ using the unit circle, we first express cotangent in terms of sine and cosine:

$ \cot(x) = \frac{\cos(x)}{\sin(x)} $

The integral becomes:

$ \int_{0}^{\frac{\pi}{4}} \cot(x) \, dx = \int_{0}^{\frac{\pi}{4}} \frac{\cos(x)}{\sin(x)} \, dx $

Let $ u = \sin(x) $. Then $ du = \cos(x) \, dx $.

Now, change the limits of integration accordingly: when $ x = 0 $, $ u = \sin(0) = 0 $, and when $ x = \frac{\pi}{4} $, $ u = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} $.

Thus, the integral becomes:

$ \int_{0}^{\frac{\sqrt{2}}{2}} \frac{1}{u} \, du = \left. \ln|u| \right|_{0}^{\frac{\sqrt{2}}{2}} $

Evaluating this, we get:

$ \ln \left( \frac{\sqrt{2}}{2} \right) – \ln(0) $

Note that $ \ln(0) $ is undefined, suggesting an improper integral. Thus, we interpret the limit at $ u \to 0^{+} $:

$ \lim_{u \to 0^{+}} \ln(u) = -\infty $

The final value of the integral is:

$ \boxed{-\infty} $

To evaluate the integral of $ cot(x) $ from $ 0 $ to $ frac{pi}{4} $ using the unit circle, we express cotangent in terms of sine and cosine:

$ int_{0}^{frac{pi}{4}} cot(x) , dx = int_{0}^{frac{pi}{4}} frac{cos(x)}{sin(x)} , dx $

Setting $ u = sin(x) $, then $ du = cos(x) , dx $.

Change the limits: $ x = 0 $ gives $ u = 0 $; $ x = frac{pi}{4} $ gives $ u = frac{sqrt{2}}{2} $.

Thus:

$ int_{0}^{frac{sqrt{2}}{2}} frac{1}{u} , du = ln left( frac{sqrt{2}}{2}
ight) – ln(0) $

The limit as $ u o 0^{+} $:

$ -infty $

To evaluate the integral of $ \cot(x) $ from $ 0 $ to $ \frac{\pi}{4} $:

$$ \int_{0}^{\frac{\pi}{4}} \cot(x) \, dx $$

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