Mathematics

Evaluate the following definite integral:
$$\displaystyle \int _{\pi /4}^{\pi /2} \cot x \ dx$$


SOLUTION
$$\displaystyle \int^{\dfrac{\pi}{2}}_{\dfrac{\pi}{4}}cotxdx$$

$$=\displaystyle \int^{\dfrac{\pi}{2}}_{\dfrac{\pi}{4}}\dfrac{cosx}{sinx}dx$$

$$=ln|sinx| ^{\dfrac{\pi}{2}}_{\dfrac{\pi}{4}}$$     $$\left[\displaystyle \because \int{\dfrac{f'(x)}{f(x)}}dx=\log|f(x)|\right]$$

$$=ln(sin\dfrac{\pi}{2})-ln\left(sin\dfrac{\pi}{4}\right)$$

$$=ln(1)-ln\left(\dfrac{1}{\sqrt{2}}\right)$$

$$=\dfrac{1}{2}ln2$$
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Subjective Medium Published on 17th 09, 2020
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