Mathematics

Evaluate: $$\displaystyle\int {\frac{{1 - \cot x}}{{1 + \cot x}}dx} $$


SOLUTION
$$I=\displaystyle\int \dfrac{1-\cot x}{1+\cot x}dx$$

$$\Rightarrow \displaystyle\int \dfrac{1-\dfrac{\cos x}{\sin x}}{1+\dfrac{\cos x}{\sin x}}dx$$

$$\Rightarrow \displaystyle\int \dfrac{\sin x-\cos x}{\sin x+\cos x}dx$$

$$\sin x+\cos x=t$$

$$(\cos x-\sin x)dx=dt$$

$$=\displaystyle\int -\dfrac{dt}{t}$$

$$=-ln |t|+c$$

$$=-ln|\sin x+\cos x|+c$$.
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Subjective Medium Published on 17th 09, 2020
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