Evaluvate the following intregals: 1 1- dx - Vidyadip

Question

Evaluvate the following intregals:
$\int\frac{1}{1-\cot\text{x}}\text{ dx}$

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Answer

Let $\text{I}=\int\frac{1}{1-\cot\text{x}}\text{ dx}$
$=\int\frac{1}{1-\frac{\cos\text{x}}{\sin\text{x}}}\ \text{dx}$
$=\int\frac{\sin\text{x}}{\sin\text{x}-\cos\text{x}}\ \text{dx}$
Let $\sin\text{x}=\lambda\frac{\text{d}}{\text{dx}}(\sin\text{x}-\cos\text{x})+\mu(\sin\text{x}-\cos\text{x})+\text{v}$
$\sin\text{x}=\lambda\frac{\text{d}}{\text{dx}}(\cos\text{x}+\sin\text{x})+\mu(\sin\text{x}-\cos\text{x})+\text{v}$
$\sin\text{x}=\cos(\lambda-\mu)+\sin\text{x}(\lambda+\mu)+\text{v}$
Compairing the cooefficients of $\sin\text{x}\ \&\cos\text{x}$ on the both the sides,
$\lambda+\mu=1\ ...(1)$
$\lambda-\mu=1\ ...(2)$
$\text{v}=0\ ...(3)$
Equation (1), (2), (3) gives
$\lambda=\frac{1}{2},\mu=\frac{1}{2},\text{v}=0$
$\text{I}=\int\frac{\frac{1}{2}(\cos\text{x}+\sin\text{x})+\frac{1}{2}(\sin\text{x}-\cos\text{x}_)}{(\sin\text{x}-\cos\text{x})}\ \text{dx}$
$=\frac{1}{2}\int\frac{(\cos\text{x}+\sin\text{x})}{(\sin\text{x}-\cos\text{x})}\ \text{dx}+\frac{1}{2}\int\ \text{dx}$
$\text{I}=\frac{1}{2}\log|\sin\text{x}-\cos\text{x}|+\frac{1}{2}\text{x}+\text{c}$

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